arXiv:1802.03045v2 [] 13 Feb 2018

ALGORITHMIC ASPECTS OF BRANCHED COVERINGS III/V.
ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT
SEQUENCE
LAURENT BARTHOLDI AND DZMITRY DUDKO
r ý be a Thurston map and let M pfrq be its mapping
Abstract. Let fr: pS 2 , Aq
r of maps obtained by pre- and post-composing fr
class biset: isotopy classes rel A
r Let A Ď A
r be an fr-invariant subset, and
by the mapping class group of pS 2 , Aq.
let f : pS 2 , Aq ý be the induced map. We give an analogue of the Birman short
r is an iterated
exact sequence: just as the mapping class group ModpS 2 , Aq
extension of ModpS 2 , Aq by fundamental groups of punctured spheres, M pfrq
is an iterated extension of M pf q by the dynamical biset of f .
Thurston equivalence of Thurston maps classically reduces to a conjugacy
problem in mapping class bisets. Our short exact sequence of mapping class
bisets allows us to reduce in polynomial time the conjugacy problem in M pfrq
to that in M pf q. In case fr is geometric (either expanding or doubly covered
by a hyperbolic torus endomorphism) we show that the dynamical biset Bpf q
r is a complete conjugacy
together with a “portrait of bisets” induced by A
r
invariant of f .
Along the way, we give a complete description of bisets of p2, 2, 2, 2q-maps
as a crossed product of bisets of torus endomorphisms by the cyclic group of
order 2, and we show that non-cyclic orbisphere bisets have no automorphism.
We finally give explicit, efficient algorithms that solve the conjugacy and
centralizer problems for bisets of expanding or torus maps.

1. Introduction
A Thurston
is a branched covering f : S 2 ý of the sphere whose post-critical
Ť map
n
set Pf :“ ně1 f pcritical points of f q is finite.
Extending [18], we developed in [2–5] an algebraic machinery that parallels the
topological theory of Thurston maps: one considers the orbisphere pS 2 , Pf , ordf q,
with ordf : Pf Ñ t2, 3, . . . , 8u defined by
ordf ppq “ l. c. m.tdegq pf n q | n ě 0, f n pqq “ pu,

and the orbisphere fundamental group G “ π1 pS 2 , Pf , ord, ˚q, which has one generator of order ordf ppq per point p P Pf , and one relation. Then f is encoded in
the structure of a G-G-biset Bpf q: a set with commuting left and right actions of
G. By [5, Theorem 7.9] (see also Corollary 3.7), the isomorphism class of Bpf q is
a complete invariant of f up to isotopy.
There is a missing element to this description, that of “extra marked points”. In
the process of a decomposition of spheres into smaller pieces, one is led to consider
Thurston maps with extra marked points, such as periodic cycles or preimages of
Date: February 8, 2018.
Partially supported by ANR grant ANR-14-ACHN-0018-01, DFG grant BA4197/6-1 and ERC
grant “HOLOGRAM”.
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LAURENT BARTHOLDI AND DZMITRY DUDKO

post-critical points. They can be added to Pf , but they have order 1 under ordf
so are invisible in π1 pS 2 , Pf , ordf q. The orbisphere orders can be made artificially
larger, but then other properties, such as the characterization of expanding maps
as those having a “contracting” biset (see [4, Theorem A]) are lost.
We resolve this issue, in this article, by introducing portraits of bisets and exhibiting their algorithmic properties. As an outcome, the conjugacy and centralizer
problems for Thurston maps with extra marked points reduces to that of the underlying map with only Pf marked.
This allows us, in particular, to understand algorithmically maps doubly covered
by torus endomorphisms.
r Ą
ordq and a sub1.1. Maps and bisets. The natural setting is an orbisphere pS 2 , A,
2
Ą
r
orbisphere pS , A, ordq; namely one has A Ď A and ordpaq | ordpaq for all a P A.
r։
r Ą
There is a corresponding morphism of fundamental groups π1 pS 2 , A,
ord, ˚q “: G
2
G :“ π1 pS , A, ord, ˚q, called a forgetful morphism. We considered in [5, §7] the
inessential case in which ordpaq ą 1 ô Ą
ordpaq ą 1, so that only the type of
singularities are changed by the forgetful functor. Here we are more interested in
r
the essential case, in which A Ř A.
In this Introduction, we restrict ourselves to self-maps of orbispheres; the general non-dynamical case is covered in §2. An orbisphere self-map f : pS 2 , A, ordq ý
is a branched covering f : S 2 ý with ordppq degp pf q | ordpf ppqq for all p P S 2 .
Given a map f : pS 2 , Aq ý, there may exist different orbisphere structures on
pS 2 , Aq turning f into an orbisphere self-map; in particular, the maximal one,
in which ordpaq “ 8 for every a P A, and the minimal one, in which ordpaq “
l. c. m.tdegpf n @xq | n ě 0, x P f ´n paqu. The self-map f is encoded by the biset
(1)

Bpf q :“ tβ : r0, 1s Ñ S 2 zA | βp0q “ ˚ “ f pβp1qqu{«A,ord ,

where «A,ord denotes homotopy rel pA, ordq, and the commuting left and right
π1 pS 2 , A, ord, ˚q-actions are given respectively by concatenation of paths or their
appropriate f -lift.
Bisets form a convenient category with products, detailed in [3]. A self-conjugacy
of a G-G-biset B is a pair of maps pφ : G ý, β : B ýq with βphbgq “ φphqβpbqφpgq;
and an automorphism of B is a self-conjugacy with φ “ 1. Cyclic bisets (that of
p t0, 8uq ý) have a special status; for the others,
maps z ÞÑ z d : pC,
Proposition A (= Corollary 4.30). If
AutpBq “ 1.

G BG

is a non-cyclic orbisphere biset then

r “ A \ tdu, obtained by adding a point to the orbisphere pS 2 , A, ordq,
Consider A
r Ą
resulting in an orbisphere pS 2 , A,
ordq. There is a short exact sequence, called the
Birman exact sequence,
(2)

r Ñ ModpS 2 , Aq Ñ 1
1 Ñ π1 pS 2 zA, dq Ñ ModpS 2 , Aq

(note that the orbisphere structure is ignored.) Let f : pS 2 , A, ordq ý be an orbisphere map, and recall that its mapping class biset is the set
M pf q :“ tm1 f m2 | m1 , m2 P ModpS 2 , Aqu{«A ,

with natural left and right actions of ModpS 2 , Aq. Assume first that the extra point
rĄ
r Ą
d is fixed by f , and write fr: pS 2 , A,
ordq ý for the map f acting on pS 2 , A,
ordq.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

3

There is then a natural map M pfrq Ñ M pf q, and we will see that it is an extension
of bisets (see Definition 2.10):
Theorem B (= Corollary 2.23). Subordinate to the exact sequence of groups (2),
there is a short exact sequence of bisets
˙
ˆ ğ
Bpf 1 q
ãÑ M pfrq ։ M pf q
1

π1 pS 2 zA,dq

f 1 PMpf q

1

2

π1 pS 2 zA,dq

where Bpf q denotes the biset of f : pS , Aq ý rel the base point d.
r “ A\tperiodic pointsu: for a cycle of
The statement can be easily extended to A
length ℓ, the fibres in the short exact sequence are of the form Bpf qℓ with left- and
right-action twisted along an ℓ-cycle. This case is at the heart of our reduction of the
conjugacy problem from M pfrq to M pf q. Approximately the same picture applies
if D contains preimages of points in A, but there are subtle complications which
are taken care of in §2, see Theorem 2.13. In essence, the presence of preperiodic
points imposes a finite index condition on centralizers, and splits conjugacy classes
into finitely many pieces, see the remark after Lemma 4.24.
1.2. Portraits of bisets. Portraits of bisets emerge from a simple remark: fixed
points of f naturally yield conjugacy classes in Bpf q. Indeed if f ppq “ p then choose
a path ℓ : r0, 1s Ñ S 2 zA from ˚ to p, and consider cp :“ ℓ#ℓ´1 Òpf P Bpf q, which
is well-defined up to conjugation by G, namely a different choice of ℓ would yield
g ´1 cp g for some g P G. Conversely, if f expands a metric, then every conjugacy
class in Bpf q corresponds to a unique repelling f -fixed point.
r ý
A portrait of bisets in Bpf q, see Definition 2.15, consists of a map f˚ : A
extending f åA ; a collection of peripheral subgroups pGa qaPAr of G: they are such that
every Ga is cyclic and generated by a “lollipop” around a; and a collection of cyclic
Ga -Gf˚ paq -bisets pBa qaPAr . Two portraits of bisets pGa , Ba qaPAr and pG1a , Ba1 qaPAr
r ý are conjugate if there exist pℓa q r in GAr such
parameterized by the same f˚ : A
aPA

1
´1 1
that Ga “ ℓ´1
a Ga ℓa and Ba “ ℓa Ba ℓf˚ paq . The set of self-conjugacies of a portrait
is called its centralizer.
r every biset admits a unique minimal portrait up to conjugacy,
In case A “ A,
which may be understood geometrically as follows. Consider a branched covering
f : pS 2 , Aq ý. For every a P A choose a small disk neighbourhood Da of it; up
to isotopy we may assume that f : Da ztau Ñ Df paq ztf paqu is a covering. A choice
of embeddings π1 pDa ztauq ãÑ π1 pS 2 , Aq yields a family pGa qaPA of peripheral subgroups; and the corresponding embeddings Bpf : Da ztau Ñ Df paq ztf paquq ãÑ Bpf q
yields a minimal portrait of bisets pGa , Ba qaPA in Bpf q.
r “ A \ te1 , . . . , en u with pe1 , . . . , en q a periodic cycle, the bisets Bei
In case A
consist of single points, and almost coincide with Ishii and Smillie’s notion of homotopy pseudo-orbits, see [13] and §4.2: imagine that pe1 , . . . , en q Ă S 2 is almost a
periodic cycle, in that f pei q is so close to ei`1 that there is a well-defined “shortest”
path ℓi from the first to the second, indices being read modulo n. Choose for each
ei
i a path mi from ˚ to ei . Set then Bi :“ tmi #pℓi`1 #m´1
i`1 qÒf u, the portrait of
bisets encoding pe1 , . . . , en q.
Theorem B is proven via portraits of bisets. There is a natural forgetful intertwiner of bisets
2 r Ą
2
r
r
(3)
r B r :“ Bpf : pS , A, ordq ýq Ñ Bpf : pS , A, ordq ýq “: G BG
G

G

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LAURENT BARTHOLDI AND DZMITRY DUDKO

r b r G. Every portrait in B,
r for example
given by b ÞÑ 1 b b b 1 where B – G bGr B
G
its minimal one, induces a portrait in B via the forgetful map. Let us denote by
r is
ModpS 2 , Aq the pure mapping class group of S 2 zA. A class m P ModpS 2 , Aq
r
called knitting if the image of m is trivial in ModpS 2 , A \ tauq for every a P AzA.
We prove:
Theorem C (= Theorem 2.19). Let G BG be an orbisphere biset with portrait
r ։ G and f˚ : A
r ý and deg : A
r Ñ N be compatible extensions.
f˚ : A ý, and let G
There is then a bijection between, on the one hand, conjugacy classes of portraits
of bisets pGa , Ba qaPAr in B parameterized by f˚ and deg and, on the other hand,
r G-bisets
r
r ։ G considered up to composition with the
Gprojecting to B under G
biset of a knitting element. This bijection maps every minimal portrait of bisets of
r to pGa , Ba q r .
B
aPA

1.3. Geometric maps. A homeomorphism f : pS 2 , Aq ý is geometric if f is either
of finite order (f n “ 1 for some n ą 0) or pseudo-Anosov (there are two transverse
measured foliations preserved by f such that one foliation is expanded by f while
another is contracted). In both cases, f preserves a geometric structure on S 2 , and
every surface homeomorphism decomposes, up to isotopy, into geometric pieces,
see [25].
Consider now a non-invertible sphere map f : pS 2 , Aq ý, and let A8 Ď A denote the forward orbit of the periodic critical points of f . The map f is Böttcher
expanding if there exists a metric on S 2 zA8 that is expanded by f , and such that
f is locally conjugate to z ÞÑ z dega pf q at every a P A8 . The map f is geometric if
f is either
tExpu Böttcher expanding; or
tGTor/2u a quotient of a torus endomorphism z ÞÑ M z`q : R2 {Z2 ý by the involution
z ÞÑ ´z, for a 2 ˆ 2 matrix M whose eigenvalues are different from ˘1.
The two cases are not mutually exclusive. A map f P tGTor/2u is expanding if
and only if the absolute values of the eigenvalues of M are greater than 1. Note
also that if f is non-invertible and covered by a torus endomorphism then either
f P tExpu or the minimal orbisphere of f satisfies #Pf “ 4 and ordf ” 2.
In that last case, we show that G BG is a crossed product of an Abelian biset
with an order-2 group. Let us fix a p2, 2, 2, 2q-orbisphere pS 2 , A, ordq and let us
set G :“ π1 pS 2 , A, ordq – Z2 ¸ t˘1u. We may identify A with the set of all
order-2 conjugacy classes of G. By Euler characteristic, every branched covering
f : pS 2 , A, ordq ý is a self-covering. Therefore, the biset of f is right principal.
We denote by Mat`
2 pZq the set of 2 ˆ 2 integer matrices M with detpM q ą 0.
2
For a matrix M P Mat`
2 pZq and a vector v P Z there is an injective endomorphism
v
2
M : Z ¸ t˘1u ý given by the following “crossed product” structure:
(4)

M v pn, 1q “ pM n, 1q and M v pn, ´1q “ pM n ` v, ´1q.

Furthermore, Z2 ¸ t˘1u has exactly 4 conjugacy classes of order 2, which we denote
by A. Then M v induces a map pM v q˚ : A ý on these conjugacy classes. We write
BM v “ BM ¸ t˘1u the crossed product decomposition of the biset of M v .
Proposition D (= Propositions 4.5 and 4.6). The biset of M v from (4) is an
orbisphere biset, and conversely every p2, 2, 2, 2q-orbisphere biset B is of the form
2
B “ BM v for some M P Mat`
2 pZq and some v P Z . Two bisets BM v and BN w

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

5

are isomorphic if and only if M “ ˘N and v ” w pmod 2Z2 q. The biset BM v is
geometric if and only if both eigenvalues of M are different from ˘1.
A distinguished property of a geometric map is rigidity: two geometric maps
are Thurston equivalent (namely, conjugate up to isotopy) if and only if they are
topologically conjugate.
An orbisphere biset G BG is geometric if it is the biset of a geometric map,
and tGTor/2u and tExpu bisets are defined similarly. If B is a geometric biset,
then by rigidity there is a map fB : pS 2 , A, ordq ý, unique up to conjugacy, with
BpfB q – B.
r r Ñ G BG is a forgetful intertwiner as in (3), then
If G BG is geometric and Gr B
G
r
elements of AzA
(which a priori do not belong to any sphere) can be interpreted
r is itself geometric,
dynamically as extra marked points on S 2 zA. More precisely, if B
2
and B is the biset of the geometric map f : pS , Aq ý, then there is an embedding
r in S 2 as an f -invariant set, unique unless G is cyclic, in such a manner that
of A
r
r ýq.
B is isomorphic to Bpf : pS 2 , Aq
Since furthermore geometric maps have only finitely many periodic points of
given period, we obtain a good understanding of conjugacy and centralizers of
geometric bisets:
r Ñ G be a forgetful morphism of groups and
Theorem E (= Theorem 4.41). Let G
let
r r Ñ G BG and r B
r1 Ñ GB1G
r BG
r
G
G G

r is geometbe two forgetful biset morphisms as in (3). Suppose furthermore that B
1
1
ric of degree ą 1. Denote by pGa , Ba qaPAr and pGa , Ba qaPAr the portraits of bisets
r and B
r 1 in B and B 1 respectively.
induced by B
1
r B
r are conjugate under ModpGq
r if and only if there exists φ P ModpGq
Then B,
such that B φ – B 1 and the portraits pGφa , Baφ qaPAr and pG1a , Ba1 qaPAr are conjugate.
Furthermore, the centralizer of the portrait pGa , Ba qaPAr is trivial, and the cenr of B
r is isomorphic, via the forgetful map ModpGq
r Ñ ModpGq, to
tralizer ZpBq
ˇ φ φ
(
φ P ZpBq ˇ pGa , Ba qaPAr „ pGa , Ba qaPAr
and is a finite-index subgroup of ZpBq.

Let us call an orbisphere map f : pS 2 , A, ordq ý weakly geometric if its minimal
quotient on pS 2 , Pf , ordf q is geometric; an orbisphere biset B is weakly geometric if
its minimal quotient orbisphere biset is geometric. In Theorem 4.35 we characterize
weakly geometric maps as those decomposing as a tuning by homeomorphisms:
starting from a geometric map, some points are blown up to disks which are mapped
to each other by homeomorphisms.
1.4. Algorithms. An essential virtue of the portraits of bisets introduced above
is that they are readily usable in algorithms. Previous articles in the series already
highlighted the algorithmic aspects of bisets; let us recall the following.
From the definition of orbisphere bisets in [5, Definition 2.6], it is clearly decidable whether a given biset H BG is an orbisphere biset; the groups G and
H may be algorithmically identified with orbisphere groups π1 pS 2 , A, ord, ˚q and
π1 pS 2 , C, ord, :q respectively, and the induced map f˚ : C Ñ A is computable. In
particular, if G BG is a G-G-biset, then the dynamical map f˚ : A ý is computable,

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LAURENT BARTHOLDI AND DZMITRY DUDKO

the minimal orbisphere quotient G :“ π1 pS 2 , Pf , ordf , ˚q is computable, and the
induced G-G-biset B is computable. It is also easy (see [4, §5]) to determine from
an orbisphere biset whether it is tGTor/2u (and then to determine an affine map
M z ` q covering it) or tExpu.
We shall show that recognizing conjugacy of portraits is decidable, and give
efficient (see below) algorithms proving it, as follows:
Algorithm 1.1 (= Algorithms 5.6 and 5.10). Given a minimal geometric orbirÑA
r of the dynamics of B on its peripheral
sphere biset G BG , an extension f˚ : A
classes, and two portraits of bisets pGa , Ba qaPAr and pG1a , Ba1 qaPAr with dynamics f˚ ,
Decide whether pGa , Ba qaPAr and pG1a , Ba1 qaPAr are conjugate, and Compute the
centralizer of pGa , Ba qaPAr , which is a finite abelian group.

Algorithm 1.2 (= Algorithms 5.7 and 5.11). Given a minimal geometric orr Ñ A
r of the dynamics of B on its
bisphere biset G BG and an extension f˚ : A
peripheral classes,
Produce a list of representatives of all conjugacy classes of portraits of bisets
pGa , Ba qaPAr in B with dynamics f˚ .

Thurston equivalence to a map f : pS 2 , Aq ý reduces to the conjugacy problem
in the mapping class biset M pf q. Let X beŤa basis of M pf q and let N be a finite
generating set of ModpS 2 , Aq; so M pf q “ ně0 N n X. We call an algorithm with
input in M pf q ˆ M pf q efficient if for f, g P N n X the running time of the algorithm
is bounded by a polynomial in n.
We deduce that conjugacy and centralizer problems are decidable for geometric
maps, as long as they are decidable on their minimal orbisphere quotients:

Corollary F (= Algorithm 5.12). There is an efficient algorithm with oracle that,
given two orbisphere maps f, g by their bisets and such that f is geometric, decides
whether f, g are conjugate, and computes the centralizer of f .
The oracle must answer, given two geometric orbisphere maps f, g on their minimal orbisphere pS 2 , Pf , ordf q respectively pS 2 , Pg , ordg q, whether they are conjugate
and what the centralizer of f is.
Algorithms for the oracle itself will be described in details in the last article of
the series [6]. Furthermore, we have the following oracle-free result, proven in §5.3:
Corollary G. There is an efficient algorithm that decides whether a rational map
is equivalent to a given twist of itself, when only extra marked points are twisted.
1.5. Historical remarks: Thurston equivalence and its complexity. The
conjugacy problem is known to be solvable in mapping class groups [11]. The
state of the art is based on the Nielsen-Thurston classification: decompose maps
along their canonical multicurve; then a complete conjugacy invariant of the map
is given by the combinatorics of the decomposition, the conjugacy classes of return
maps, and rotation parameters along the multicurve. For general surfaces, the
cost of computing the decomposition is at most exponential time in n, see [14, 24],
and so is the cost of comparing the pseudo-Anosov return maps [17]. Margalit,
Yurttas, Strenner recently announced polynomial-time algorithms for all the above.
At all rates, for punctured spheres the cost of computing the decomposition is
polynomial [7].

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

7

Kevin Pilgrim developed, in [20], a theory of decompositions of Thurston maps
extending the Nielsen-Thurston decomposition of homeomorphisms. It is a fundamental ingredient in understanding Thurston equivalence of maps, without any
claims on complexity.
In [1], a general strategy is developed, along with computations of the mapping
class biset for the three degree-2 polynomials with three finite post-critical points
and period respectively 1, 2, 3. The Douady-Hubbard “twisted rabbit problem” is
solved in this manner (it asks to determine for n P Z the conjugacy class of r ˝ tn
with rpzq « z 2 ´ 0.12256 ` 0.74486i the “rabbit” polynomial and tpzq the Dehn
twist about the rabbit’s ears). Russell Lodge computed in [16] the solution to an
array of other “twisted rabbit” problems, by finding explicitly the mapping class
biset structure.
If M pf q is a contracting biset, then its conjugacy problem may be solved in
quasi-linear time. In terms of a twist parameter such as the n above, the solution
has Oplog nq complexity. However, this bound is not uniform, in that it requires
e.g. the computation of the nucleus of M pf q, which cannot a priori be bounded.
Nekrashevych showed in [19, Theorem 7.2] that the mapping class biset of a hyperbolic polynomial is contracting. Conversely, if M pf q contains an obstructed map
then it is not contracting.
In the case of polynomials, however, even more is possible: bisets of polynomials admit a particularly nice form by placing the basepoint close to infinity (this
description goes hand-in-hand with Poirier’s “supporting rays” [21]). As a consequence, the “spider algorithm” of Hubbard and Schleicher [12] can be implemented
directly at the level of bisets and yields an efficient algorithm, also in practice since
it does not require the computation of M pf q’s nucleus. This algorithm will be
described in [6].
Selinger and Yampolsky showed in [23] that the canonical decomposition is computable. In this manner, they solve the conjugacy problem for maps whose canonical
decomposition has only rational maps with hyperbolic orbifold: a complete conjugacy invariant of the map is given by the combinatorics of the decomposition, the
conjugacy classes of its return maps, and rotation parameters along the canonical
obstruction.
We showed in [5] that the conjugacy problem is decidable in general: a complete
conjugacy invariant of a Thurston map is given by the combinatorics of the decomposition, together with conjugacy classes of return maps, rotation parameters
along the canonical obstruction, together with the induced action of the centralizer
groups of return maps.
We finally mention a different path towards understanding Thurston maps, in
the case of maps with four post-critical points: the “nearly Euclidean Thurston
maps” from [8]. There, the restriction on the size of the post-critical set implies
that the maps may be efficiently encoded via linear algebra; and as a consequence,
conjugacy of NET maps is efficiently decidable. We are not aware of any direct
connections between their work and ours.
1.6. Notations. Throughout the text, some letters keep the same meaning and are
not always repeated in statements. The symbols A, C, D, E denote finite subsets of
the topological sphere S 2 . There is a sphere map fr: pS 2 , C \ Eq Ñ pS 2 , A \ Dq,
which restricts to a sphere map f : pS 2 , Cq Ñ pS 2 , Aq. Implicit in the definition, we
r “ C \E and A
r “ A\D. If there
have f pCqYtcritical values of f u Ď A. We write C

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LAURENT BARTHOLDI AND DZMITRY DUDKO

are, furthermore, orbisphere structures on the involved spheres, we denote them by
pS 2 , A, ordq etc., with the same symbol ‘ord’. We also abbreviate X :“ pS 2 , A, ordq
and Y :“ pS 2 , C, ordq.
For sphere maps f0 , f1 : pS 2 , Cq Ñ pS 2 , Aq, we mean by f0 «C f1 that f0 and f1
are isotopic, namely there is a path pft qtPr0,1s of sphere maps ft : pS 2 , Cq Ñ pS 2 , Aq
connecting them.
For γ a path and f a (branched) covering, we denote by γÒxf the unique f -lift of
γ that starts at the preimage x of γp0q.
We denote by f åZ the restriction of a function f to a subset Z of its domain.
Finally, for a set Z we denote by Z Ó. the group of all permutations of Z.
2. Forgetful maps
We recall a minimal amount of information from [5]: a marked orbisphere is
pS 2 , A, ordq for a finite subset A Ă S 2 and a map ord : A Ñ t2, 3, . . . , 8u, extended to S 2 by ordpS 2 zAq ” 1. For a choice of basepoint ˚ P S 2 zA, its orbispace
fundamental group is generated by “lollipop” loops pγa qaPA based at ˚ that each
encircle once counterclockwise a single point of A, and with A “ ta1 , . . . , an u has
presentation
(5)

nq
1q
, γa1 ¨ ¨ ¨ γan y.
, . . . , γaordpa
G “ π1 pS 2 , A, ord, ˚q “ xγa1 , . . . , γan | γaordpa
n
1

Abstractly, i.e. without reference to a sphere, an orbisphere group is a group G as
in (5) together with the conjugacy classes Γ1 , . . . , Γn of γa1 , . . . , γan respectively.
An orbisphere map f : pS 2 , C, ordC q Ñ pS 2 , A, ordA q between orbispheres is
an orientation-preserving branched covering between the underlying spheres, with
f pCq Y tcritical values of f u Ď A, and with ordC ppq degp pf q | ordA pf ppqq for all
p P S2.
To avoid special cases, we make, throughout this article except in §6.3, the assumption
(6)

#A ě 3

and

#C ě 3.

For #A “ 2, many things go wrong: one must require ord to be constant; the fundamental group has a non-trivial centre; and the degree-d self-covering of pS 2 , A, ordq
has an extra symmetry of order d ´ 1. All our statements can be modified to take
into account this special case, see §6.3.
Fix basepoints ˚ P S 2 zA and : P S 2 zC. The orbisphere biset of an orbisphere
map f : pS 2 , C, ordC q Ñ pS 2 , A, ordA q is the π1 pS 2 , C, ordC , :q-π1 pS 2 , A, ordA , ˚qbiset
Bpf q “ tβ : r0, 1s Ñ S 2 zC | βp0q “ :, f pβp1qq “ ˚u { «C,ordC ,
with ‘«C,ord ’ denoting homotopy in the orbispace pS 2 , C, ordq. An orbisphere biset
H BG can also be defined purely algebraically, see [5, §7 and Definition 2.6]. By
[5, Theorem 7.9], there is an orbisphere map f : pS 2 , C, ordC q Ñ pS 2 , A, ordA q,
unique up to isotopy, such that B is isomorphic to Bpf q. We denote by B˚ : C Ñ A
the induced map on the peripheral conjugacy classes of H and G – they are identified
with the associated punctures.
Ą be orbispheres, and suppose ordpaq | Ą
Let pS 2 , A, ordq and pS 2 , A \ D, ordq
ordpaq
for all a P A. We then have a natural forgetful homomorphism FD : π1 pS 2 , A \
D, Ą
ord, ˚q Ñ π1 pS 2 , A, ord, ˚q given by γa ÞÑ γa for a P A and γd ÞÑ 1 for d P D.
We write the forgetful map pS 2 , A \ D, Ą
ordq 99K pS 2 , A, ordq with a dashed arrow,

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

9

because even though FD is a genuine group homomorphism, the corresponding map
between orbispheres is only densely defined. Note, however, that its inverse is a
genuine orbisphere map.
Consider forgetful maps pS 2 , C \E, Ą
ordq 99K pS 2 , C, ordq and pS 2 , A\D, Ą
ordq 99K
2
pS , A, ordq; in the sequel we shall keep the notations
X :“ pS 2 , A, ordq,

r :“ A \ D,
A

Y :“ pS 2 , C, ordq,

r :“ C \ E.
C

Ą Ñ pS 2 , A,
r ordq
r Ą
Let fr: pS 2 , C,
ordq be an orbispace map, with such that fr restricts
to an orbisphere map f : pS 2 , C, ordq Ñ pS 2 , A, ordq. We thus have
Ą
r ordq
pS 2 , C,

(7)

FE

X “ pS 2 , C, ordq

fr

f

rĄ
pS 2 , A,
ordq
FD

pS 2 , A, ordq “ Y.

(In particular, frpCq Ď A and A contains all critical values of fr.) We are concerned,
in this section, with the relationship between Bpfrq and Bpf q; we shall show that
Bpfrq may be encoded by Bpf q and a portrait of bisets, and that this encoding is
unique up to a certain equivalence. The algebraic counterpart of (7) is
(8)

Ą ˚q “: G
r
r Ą
r ord,
r :“ π1 pS 2 , C,
H
ord, :q í Bpfrq ì π1 pS 2 , A,
FE

FE,D

FD

H :“ π1 pS 2 , C, ord, :q í Bpf q ì π1 pS 2 , A, ord, ˚q “: G,
where we denote by
(9)

r rG
FE,D : Bpfrq Ñ Bpf q – H bH
Ă Bpf q bG

the natural map given by b ÞÑ 1 b b b 1.

2.1. Braid groups and knitting equivalence. We recall that there is no difference between ModpS 2 , Aq and ModpXq, see [5, §7]. We shall define a subgroup
r which
ModpX|Dq intermediate between ModpXq – ModpS 2 , Aq and ModpS 2 , Aq
r
will be useful to relate Bpf q and Bpf q.

Definition 2.1 (Pure braid group). Let D be a finite set on a finitely punctured
sphere S 2 zA. The pure braid group BraidpS 2 zA, Dq is the set of continuous motions
m : r0, 1s ˆ D Ñ S 2 zA considered up to isotopy so that
‚ mpt, ´q : D ãÑ S 2 zA is an inclusion for every t P r0, 1s;
‚ mp0, ´q “ mp1, ´q “ 1 åD .
The product in BraidpS 2 zA, Dq is concatenation of motions, and the inverse is
time reversal.
△
Note that in the special case D “ t˚u we have BraidpS 2 zA, t˚uq “ π1 pS 2 zA, ˚q.
Theorem 2.2 (Birman). For every m P BraidpS 2 zA, Dq there is a unique mapping
class pushpmq P kerpModpS 2 , A\Dq Ñ ModpS 2 , Aqq such that pushpmq is isotopic
rel A to the identity via an isotopy moving D along m´1 . The map
(10)

push : BraidpS 2 zA, Dq Ñ kerpModpS 2 , A \ Dq Ñ ModpS 2 , Aqq

is an isomorphism (it would be merely an epimorphism if #A ď 2).



10

LAURENT BARTHOLDI AND DZMITRY DUDKO

From now on we identify BraidpS 2 zA, Dq with its image under (10).
Definition 2.3 (Knitting group). Let X “ pS 2 , A, ordq be an orbisphere and let D
be a finite subset of S 2 zA. The knitting braid group knBraidpX, Dq is the kernel
of the forgetful morphism
ź
ED : BraidpS 2 zA, Dq Ñ
π1 pX, dq;
dPD

2

it is the set of D-strand braids in S zA all of whose strands are homotopically
trivial in X.
△
In case X “ pS 2 , Aq, knitting elements are the “p#D ´ 1q-decomposable braids”
from [15].
Lemma 2.4. The knitting group knBraidpX, Dq is a normal subgroup of ModpS 2 , A \ Dq.
Proof. We show that for every m P knBraidpX, Dq and h P ModpS 2 , A \ Dq
we have h´1 mh P knBraidpX, Dq. Indeed, h restricts to an orbisphere map
h : X ý fixing D pointwise. Thus mpd, ´q is a trivial loop in π1 pX, dq if and
only if hpmpd, ´qq is a trivial loop in π1 pX, dq for every d P D.

Define
(11)

ModpX|Dq :“ ModpS 2 , A \ Dq{knBraidpX, Dq.

With G “ π1 pX, ˚q, we write ModpG|Dq “ ModpX|Dq and ModpGq “ ModpXq.
We interpret elements of ModpGq as outer automorphisms of G, as mapping classes
and as biprincipal bisets. We also introduce the notation
ź
π1 pX, Dq :“
π1 pX, dq – GD .
dPD

Note the following four exact sequences:
knBraidpX, Dq

BraidpS 2 zA, Dq
ED

(12)

r “ ModpS 2 , A \ Dq
ModpGq

π1 pX, Dq;
push

ModpX|Dq

FD
FD

ModpGq “ ModpS 2 , Aq
exactness follows by definition except surjectivity in the top sequence. Given a
sequence of loops pγd qdPD P π1 pX, Dq, we may isotope them slightly to obtain
γd ptq ‰ γd1 ptq for all t P r0, 1s and all d ‰ d1 . Then bpd, tq :“ γd ptq is a braid in
S 2 zA and defines via ‘push’ an element of BraidpS 2 zA, Dq mapping to pγd q.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

11

2.2. fr-impure mapping class groups. As in (7), consider an orbisphere map
fr: pS 2 , C\E, Ą
ordq Ñ pS 2 , A\D, Ą
ordq that projects to f : pS 2 , C, ordq Ñ pS 2 , A, ordq.
We will enlarge the groups in (12) to “fr-impure mapping class groups” so that exact
sequences analogous to (16) hold.
Let Mod˚ pS 2 , C \ Eq be the group of homeomorphisms m : pS 2 , C \ Eq ý
considered up to isotopy rel C \ E such that m åC is the identity and for every
e P E we have frpmpeqq “ frpeq; i.e., m may permute points in fr´1 pfrpeqqzC. There
is a natural forgetful morphism
FE : Mod˚ pS 2 , C \ Eq Ñ ModpS 2 , Cq.

As in Definition 2.1, the braid group Braid˚ pS 2 zC, Eq is the set of continuous
motions m : r0, 1s ˆ E Ñ S 2 zC considered up to isotopy so that
‚ mpt, ´q : E ãÑ S 2 zC is an inclusion for every t P r0, 1s;
‚ mp0, ´q “ 1 åE ;
‚ mp1, eq P fr´1 pfrpeqq for every e P E.

Every m P Braid˚ pS 2 zC, Eq induces a permutation πm : e ÞÑ mp1, eq of E. The
pure braid group consists of those permutations with πm “ 1. The product in
Braid˚ pS 2 zC, Eq is m ¨ m1 “ m#pm1 ˝ p1 ˆ πm qq, with as usual ‘#’ standing
for concatenation of motions. Birman’s theorem (a slight generalization of Theorem 2.2) still holds: the group Braid˚ pS 2 zC, Eq is isomorphic to the kernel of
Mod˚ pS 2 , C \ Eq Ñ ModpS 2 , Cq via the push operator.
Let π1˚ pY, Eq be the group of motions m : r0, 1s ˆ E Ñ Y “ pS 2 , C, ordq, considered up to homotopy, such that
‚ mp0, ´q “ 1 åE ;
‚ mp1, eq P fr´1 pfrpeqq for every e P E;

here m, m1 : E ãÑ Y are homotopic if mp´, eq and m1 p´, eq are homotopic curves
(relative to their endpoints) in Y for all e P E. The product in π1˚ pY, Eq is again
m ¨ m1 “ m#pm1 ˝ p1 ˆ πm qq. We have
ź
ź
pfr´1 pdq X EqÓ.,
(13)
π1˚ pY, Eq –
π1 pY, eq ¸
ePE

dPfrpEq

the isomorphism mapping m to its restrictions mp´, eq and its permutation πm .

Lemma 2.5. The following sequence is exact:
(14)

knBraidpY, Eq

Braid˚ pS 2 zC, Eq

EE

π1˚ pY, Eq,

where EE is the natural forgetful morphism.
Proof. Suppose EE pmq “ EE pm1 q. Then m´1 m1 is pure so m´1 m1 P knBraidpY, Eq.
The converse is also obvious.

The same argument as in Lemma 2.4 shows that the knitting group knBraidpY, Eq
is a normal subgroup of Mod˚ pS 2 , C \ Eq. We may thus define
(15)

Mod˚ pY |Eq :“ Mod˚ pS 2 , C \ Eq{knBraidpY, Eq.

12

LAURENT BARTHOLDI AND DZMITRY DUDKO

As in (12) we have the following exact sequences
knBraidpY, Eq

Braid˚ pS 2 zC, Eq
EE

(16)

Mod˚ pS 2 , C \ Eq

π1˚ pY, Eq.
push

Mod˚ pY |Eq

FE
FE

ModpHq “ ModpS 2 , Cq
2.3. Branched coverings. Recall that, for orbisphere maps f0 , f1 : pS 2 , C, ordq Ñ
pS 2 , A, ordq we write f0 «C f1 , and call them isotopic, if there is a path pft qtPr0,1s
of orbisphere maps ft : pS 2 , C, ordq Ñ pS 2 , A, ordq. Equivalently,
Lemma 2.6. f0 «C f1 if and only if hf0 “ f1 for a homeomorphism h : pS 2 , Cq ý
that is isotopic to the identity.
Proof. If there exists an isotopy pht q witnessing 1 «C h, then ft :“ ht f0 witnesses
f0 «C f1 . Conversely, since all critical values of ft are frozen in A, the set ft´1 pyq
moves homeomorphically for every y P S 2 (equivalently, no critical points collide).
Therefore, we may factor ft “ ht f0 , with ht pzq the trajectory of z P ft´1 pyq; this
defines an isotopy from 1 to h :“ h1 .

Consider orbisphere maps fr, r
g : pS 2 , C \ E, Ą
ordq Ñ pS 2 , A\ D, Ą
ordq as in (7). We
write fr «C|E r
g, and call fr, r
g knitting-equivalent, if fr “ hr
g for a homeomorphism
h : pS 2 , C \ Eq ý in knBraidpY, Eq; we have
fr «C\E r
g ùñ fr «C|E gr ùñ fr «C r
g.

For m P BraidpS 2 zA, Dq we define its pullback pfrq˚ m : r0, 1s ˆ E Ñ S 2 zC by
#
mp´, frpeqqÒefr if frpeq P D,
˚
ppfrq mqp´, eq :“
e
if frpeq P A.

This defines a motion of E; note that pfrq˚ mp1, eq need not equal e:

Lemma 2.7. If pushpmq P BraidpS 2 zA, Dq, then pushppfrq˚ mq defines an element
of Braid˚ pS 2 zC, Eq and we have the following commutative diagram:
(17)

pS 2 , C \ Eq
fr

pS 2 , A \ Dq

pushppfrq˚ mq

pS 2 , C \ Eq

pushpmq

pS 2 , A \ Dq.

«C\E

fr

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

13

Proof. Let us discuss in more detail the operator ‘push’. Consider a simple arc
γ : r0, 1s Ñ S 2 zA and let U Ă S 2 zA be a small disk neighborhood of γ. We can
define (in a non-unique way) a homeomorphism pushpγq : pS 2 , Aq ý that maps γp0q
to γp1q and is identity away from U. Let U1 , U2 , . . . , Ud be the preimages of U under
fr, where d “ degpfrq. Each Ui contains a preimage γi of γ. Let pushpγi q : Ui ý be
the lift of pushpγq åU under fr: Ui Ñ U, extended by the identity on S 2 zUi . Then
(18)

pushpγ1 qpushpγ2 q ¨ ¨ ¨ pushpγd q ¨ fr “ fr ¨ pushpγq.

For m P BraidpS 2 zA, Dq, we can define pushpmq as a composition of pushes
along finitely many simple arcs βi . Using (18) we lift all pushpβi q through fr;
considering the equation rel C \ E we obtain (17).


We note that pushpmq does not necessarily lift to a ‘push’ if m moves a critical
value. Indeed if γ is a simple loop then pushpγq is the quotient of two Dehn twists
about the boundary curves of an annulus surrounding γ, see [9, §4.2.2]; however
the lift of γ will not be a union of simple closed curves if γ contains a critical value;
an annulus around γ will not lift to an annulus, but rather to a more complicated
surface F ; and the quotient of Dehn twists about boundary components of F will
not be a quotient of Dehn twists about boundary components of annuli.

Proposition 2.8. Let fr: pS 2 , C \ E, Ą
ordq Ñ pS 2 , A \ D, Ą
ordq be an orbisphere map
as in (7). Then every element in knBraidpX, Dq lifts through fr to an element of
knBraidpY, Eq.
If gr : pS 2 , C \ E, Ą
ordq Ñ pS 2 , A\ D, Ą
ordq is another orbisphere map, then fr «C|E
g if and only if fr «C hr
r
gk for some h P knBraidpY, Eq, k P knBraidpX, Dq.

Proof. Consider h P knBraidpX, Dq. By Theorem 2.2 we may write h “ pushpbq.
Since bp´, dq is homotopically trivial in X for every d P D, the curve bp´, frpeqqÒefr
ends at e for all e P E with frpeq P D, because this curve is in Y . Therefore,
pfrq˚ bp1, ´q “ 1 åE , and pushppfrq˚ bq P BraidpS 2 zC, Eq. Since the lifts bp´, frpeqqÒer
f

are homotopically trivial in Y for all e P E, we have pushppfrq˚ bq P knBraidpY, Eq
and Lemma 2.7 concludes the first claim.
The second claim is a direct consequence of the first.


As a consequence, we may detail a little bit more the map ED in (12). Choose
for every d P D a path ℓd in S 2 zA from ˚ to d. This path defines an isomorphism
π1 pX, dq Ñ π1 pX, ˚q “ G by γ ÞÑ ℓd #γ#ℓ´1
d . We thus have a map
(19)

ED : BraidpS 2 zA, Dq ։ GD ,

m ÞÑ pℓd #ppush´1 pmq åd q#ℓ´1
d qdPD ,

and kerpED q “ knBraidpX, Dq.
2.4. Mapping class bisets. We introduce some notation parallel to that in (12)
r Ñ H and FD : G
r Ñ G be forgetful
and (16) for mapping class bisets. Let FE : H
r
morphisms of orbisphere groups as in (8), and let B be an orbisphere biset. Let
r r G be the induced H-G-biset. We have forgetful morphisms of
B :“ H bH
Ă B bG
r Ñ ModpGq and FE : ModpHq
r Ñ ModpHq. Corresponding
groups FD : ModpGq
mapping class bisets are written respectively, with fr and f the orbisphere maps

14

LAURENT BARTHOLDI AND DZMITRY DUDKO

r and B,
associated with B

r “ M pfrq – tn b B
r b m | n P ModpHq,
r m P ModpGqu{–
r
M pBq

r m P ModpS 2 , Aqu{«
r
“ tnfrm | n P ModpS 2 , Cq,
r,
C

M pBq “ M pf q – tn b B b m | n P ModpHq, m P ModpGqu{–

“ tnf m | n P ModpS 2 , Cq, m P ModpS 2 , Aqu{«C

together with the natural forgetful intertwiner
(20)

r Ñ M pBq,
FE,D : M pBq

r 1 Ñ FE,D pB 1 q “ H b Ă B
r 1 b r G.
B
G
H

We may also define the following mapping class biset, sometimes larger than
r assume first that Ą
M pBq:
ord is constant on E, possibly 8, and set
(21)
ˇ
+
#
ˇ
r 1 q P M pBq
F
p
B
ˇ
E,D
1
˚ r
˚ r
r an Hr G-orbisphere
r
biset ˇ
{–,
M pBq “ M pf q :“ B
r 1 q˚ “ B
r˚
ˇ
pB

r 1 q˚ : C
rÑA
r denotes the induced map on marked conjugacy classes. It is
where pB
˚
2 r
r
r 1 P M ˚ pBq
r for
an Mod pS , Cq-ModpS 2 , Aq-biset;
note indeed that we have n b B
˚
1
˚ r
2 r
r
r
B P M pBq and n P Mod pS , Cq, because FE,D pn b Bq “ FE pnq b B. Again
there is a natural forgetful intertwiner
(22)

FE,D :

r M
Mod˚ pS 2 ,Cq

˚

pfrqModpS 2 ,Aq
r Ñ ModpS 2 ,Cq M pf qModpS 2 ,Aq .

r on M pfrq does not necessarily extend
We note that the left action of ModpS 2 , Cq
˚
2 r
to an action of Mod pS , Cq on M pfrq, because the result of the action is in general
in M ˚ pf q and not in M pf q, see Example 6.2.
In case Ą
ord is not constant on E, we should be careful, because permutation
r invariant; rather, the image of H
r under such a
of points in E does not leave H
r
r and
permutation gives an orbisphere group isomorphic to H. However, M ˚ pBq
˚
2 r
Mod pS , Cq do not depend on the orbisphere structure, so the definition may be
r and G
r replaced by orbisphere groups with larger orders.
applied with H
Let us call the set of extra marked points E saturated if
fr´1 pfrpEqq Ď C \ E.

Lemma 2.9.
(1) The mapping class biset M ˚ pfrq is left-free.
(2) Suppose that E is saturated and that g0 , g1 : pS 2 , C \ E, Ą
ordq Ñ pS 2 , A \
Ą
D, ordq are orbisphere maps coinciding on C \ E and such that FE,D pg0 q
and FE,D pg1 q are isotopic through maps pS 2 , Cq Ñ pS 2 , Aq. Then g0 , g1 P
M ˚ pfrq, and there is an m P Braid˚ pS 2 zC, Eq such that mg0 “ g1 holds in
M ˚ pfrq.
(3) If E is saturated, then

(23)

M ˚ pfrq “ tmfrn | m P Mod˚ pS 2 , C \ Eq, n P ModpS 2 , A \ Dqu{«Cr .

Proof. The proof of the first claim follows the lines of [5, Proposition 6.4]: suppose
r Ñ pS 2 , Aq
r for some m P Mod˚ pS 2 , C\
that g, mg are isotopic through maps pS 2 , Cq
˚ r
Eq and g P M pf q. By Lemma 2.6, we may assume g “ mg as maps; then the
homeomorphism m is a deck transformation of the covering induced by g, so m has

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

15

finite order because degpgq ă 8. Recall that #C ě 3 by our standing assumption (6). Since m fixes at least 3 points in C and m has finite order, we deduce
that m is the identity. This shows that M ˚ pfrq is left-free.
For the second claim, let pgt : pS 2 , Cq Ñ pS 2 , AqqtPr0,1s be an isotopy between g0
and g1 . By Lemma 2.6, we may write gt at mt g0 for mt : pS 2 , Cq ý. Then m1
preserves E because E is saturated, so m1 P Braid˚ pS 2 zC, Eq as required.
The third claim directly follows from the second.

Definition 2.10 (Extensions of bisets, see [5, Definition 5.2]). Let G1 BG2 be a
G1 -G2 -biset and let N1 , N2 be normal subgroups of G1 and G2 respectively, so that
for i “ 1, 2 we have short exact sequences
(24)

1

Ni

π

Gi

Qi

1.

If the quotient Q1 -Q2 -biset N1 zB{N2 , consisting of connected components of N1 BN2 ,
is left-free, then the sequence
(25)

N 1 BN 2

G 1 BG 2

π

Q1 pN1 zB{N2 qQ2

is called an extension of left-free bisets.

△

r ։ H and G
r ։ G be surjective
Definition 2.11 (Inert biset morphism). Let H
r
r
r
group homomorphisms, and let B be a left-free H-G-biset. Recall that the tenr r G is isomorphic to the double quotient kerpH
r Ñ
sor product B :“ H bH
Ă B bG
r kerpG
r Ñ Gq with natural H-G-actions. The natural map F : B
r Ñ B is
HqzB{
r
called inert if B is a left-free biset and the natural map t¨u bGr B Ñ t¨u bG B is a
r In other words,
bijection. In particular, B has the same number of left orbits as B.
r
r
assuming that the groups H and H have similarly-written generators and so do G
r
and G, the wreath recursions of B and B are identical.
rker ãÑ ĂB
r r ։ H BG the kernel is a
Yet said differently, in the extension ker B
H G
r“H
r and G “ H so that the bisets can
disjoint union of left-principal bisets. If G
r
rÑGÑ
be iterated, then B Ñ B is inert precisely when we have a factorization G
r
IMGGr pBq, the latter groupŮbeing the quotient of G by the kernel of the right action
on the rooted tree t¨u bH ně0 B bn , see [5, §7.3].
△
Define

r
M ˚ pB|E, Dq :“ knBraidpY, EqzM ˚ pBq{knBraidpX,
Dq;

(26)

this is naturally a Mod˚ pY |Eq-ModpX|Dq-biset, and Proposition 2.8 implies in
particular that it is left-free:
Proposition 2.12. The natural forgetful maps
r
r Ñ
Ă M pBq
ModpHq

ModpGq

ModpH|Eq M pB|E, DqModpG|Dq

and

Mod˚ pS 2 ,C\Eq M

are inert.

˚
E Ó.

˚

˚
r
pBq
ModpS 2 ,A\Dq Ñ Mod˚ pY |Eq M pB|E, DqModpX|Dq



Let
denote the group of all permutations t : E ý such that frptpeqq “ frpeq.
˚
˚
We denote by H E ¸E Ó. the semidirect product where E Ó. acts on H E by permuting
˚
coordinates; compare with (13). We have π1˚ pY, Eq – H E ¸ E Ó. .

16

LAURENT BARTHOLDI AND DZMITRY DUDKO

We denote by

Braid˚ pY,Eq M
˚ r
˚

˚

r
pBq
BraidpX,Dq and

π1˚ pY,Eq M

˚

pB|E, Dqπ1 pX,Dq the

restrictions of M pBq and M pB|E, Dq to braid and fundamental groups.

Theorem 2.13. If E is saturated, then the following sequences are extensions of
bisets:
(27)
r
M ˚ pBq
knBraidpY,Eq

knBraidpX,Dq

Braid˚ pY,Eq M

˚

r
pBq
BraidpX,Dq

EE,D

r
M ˚ pBq
FE,D

π1˚ pY,Eq M

pB|E, Dqπ1 pX,Dq
P

˚

M pB|E, Dq

FE,D

M pBq

˚

˚

H E ¸E Ó.

!

–

pB 1 P M pBq, pBc1 qcPCr q
pBc1 qcPCr is a portrait in B 1

)

.
GD

(The isomorphism on the right is the topic of Theorem 2.19, and will be proven
there.)

´1
r
pBq is a connected subbiset of Braid˚ pY,Eq M ˚ pBq
Proof. By Lemma 2.9(2), FE,D
BraidpX,Dq ;
thus the central-to-left sequence is an extension of bisets. Exactness of other se
quences follows from Proposition 2.12.

r by M pBq
r in (27),
Note that, if E were not saturated or if we replaced M ˚ pBq
then we wouldn’t have exact sequences of bisets anymore, because the fibres of
r ։ M pBq wouldn’t have to be connected; see Example 6.1. The failure of
M ˚ pBq
transitivity is described by Lemma 2.20. There are similar exact sequences in case
f˚ : E Ñ D is a bijection, see Theorem 2.22.
2.5. Portraits of bisets. First, a portrait of groups amounts to a choice of representative in each peripheral conjugacy class of an orbisphere group:
Definition 2.14 (Portraits of groups). Let G be an orbisphere group with marked
r be a finite set containing A. A portrait of
conjugacy classes pΓa qaPA and let A
groups pGa qaPAr in G is a collection of cyclic subgroups Ga ď G so that
#
xgy for some g P Γa , if a P A,
Ga “
x1y otherwise.
r “ A, then pGa qaPA is called a minimal portrait of groups.
If A

△

Definition 2.15 (Portraits of bisets). Let H, G be orbisphere groups with peripheral classes indexed by C, A respectively, let H BG be an orbisphere biset, and let
f˚ : C Ñ A be its portrait. We also have a “degree” map deg : C Ñ N. A portrait
of bisets relative to these data consists of

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

17

‚ portraits of groups pHc qcPCr in H and pGa qaPAr in G;
rÑA
r of f˚ and deg : C
r Ñ N of deg;
‚ extensions f˚ : C
‚ a collection pBc qcPCr of subbisets of B such that every Bc is an Hc -Gf˚ pcq biset that is right-transitive and left-free of degree degpcq, and such that if
f˚ pcq “ f˚ pc1 q and HBc “ HBc1 qua subsets of B then c “ c1 .
r “ C and A
r “ A, then pBc qcPC is called a minimal portrait of bisets.
If C
△

r
Note in particular that if c P CzC
then Hc is trivial and the subbiset Bc consists
of degpcq elements permuted by Gf˚ pcq . If moreover f˚ pcq R A, then degpcq “ 1.
For simplicity the portrait of bisets is sometimes simply written pBc qcPCr , the other
data f˚ , deg, pHc qcPCr , pGa qaPAr being implicit.
Here is a brief motivation for the definition. Consider an expanding Thurston
map f and its associated biset B. Recall from [18] that bounded sequences in B
represent points in the Julia set of f ; in particular constant sequences represent
fixed points of f and vice versa. On the other hand, fixed Fatou components
of f are represented by local subbisets of B with same cyclic group acting on
the left and on the right. All of these are instances of portraits of bisets in B.
Furthermore, (pre)periodic Julia or Fatou points are represented by portraits of
bisets with (pre)periodic map f˚ . E.g., closures of Fatou components intersect if
and only if they admit portraits that intersect; and similarly for inclusion in the
closure of a Fatou component of a (pre)periodic point in the Julia set.
r Ñ G be a marked forgetful morphism of orbisphere groups as in (8).
Let FD : G
r
For all a P A choose a small disk neighbourhood Ua Q a that avoids all other points
r so that π1 pUa zAq
r – Z. We call a curve γ close to a if γ Ă Ua .
in A,
r a q r in G
r is uniquely characLemma 2.16. Every minimal portrait of groups pG
aPA
terized by a family of paths pℓa qaPAr with
(28)

r
ℓa : r0, 1s Ñ S 2 rel A,

r for t ă 1, ℓa p0q “ ˚ and ℓa p1q “ a,
ℓa ptq R A

so that for any sufficiently small ǫ ą 0
(29)

r a “ ℓa år0,1´ǫs #αa #pℓa år0,1´ǫs q´1 | αa is close to a
G

(

r
rel A.

Conversely, every collection of curves (28) defines by (29) a minimal portrait of
groups for every sufficiently small ǫ ą 0.
Proof. This follows immediately from the definition of “lollipop” generators, see (5).

ra is self-normalizing in G:
r conjugating G
ra by an element
It follows that every G
r
g R Ga amounts to precomposing the path ℓa with g, resulting in a different path.

Lemma 2.17. Let H BG be an orbisphere biset. Then every pair of minimal portraits of groups pHc qcPC in H and pGa qaPA in G can be uniquely completed to a
minimal portrait of bisets pBc qcPC in B.

As a consequence, the minimal portrait of bisets is unique up to conjugacy. We
note that the lemma is also true in case H BG is a cyclic biset, namely if H, G – Z;
in this case all Bc are equal to B.
Proof. Since by Assumption (6) the sets A and C contain at least 3 elements, H
and G are non-cyclic, and in particular have trivial centre.

18

LAURENT BARTHOLDI AND DZMITRY DUDKO

Let f˚ : C Ñ A be the portrait of B. Choose generators ga P Ga and hc P Hc .
Recall from [3, §2.6] or [5, Definition 2.5] that there are bc P B and kc P H such
degpcq
that kc´1 hc kc bc “ bc gf pcq for all c P C. Set then Bc :“ Hc kc bc Gf pcq , and note that
these subbisets satisfy Definition 2.15.
Suppose next that pBc1 qcPC is another portrait of bisets, and choose elements
1
bc P Bc1 . Then by [5, Definition 2.6(SB3 )] the conjugacy class ∆c appears exactly
once among the lifts of Γf pcq , so Hb1c X Bc ‰ H, so we may choose kc P H with
kc b1c P Bc . Then kc Bc1 “ kc b1c Gf pcq “ Bc Gf pcq “ Bc . We have kc Hc “ Hc kc , so
kc P Hc because Hc is self-normalizing in H, and therefore Bc “ Bc1 .

r r Ñ H BG of orbisphere bisets,
Consider next a forgetful morphism FE,D : H
Ă BG
rc q r be the minimal portrait of bisets given by Lemma 2.17.
and let pB
cPC

Lemma 2.18. Let pmc qcPCr and pℓa qaPAr be the paths (see Lemma 2.16) associr c q r and pG
ra q r . The portrait
ated with the respective portraits of bisets pH
cPC
aPA
rc qq r admits the following description: for every sufficiently small ǫ ą 0,
pFE,D pB
cPC
!
)
β p1q
(30)
Bc “ mc år0,1´ǫs #βc #pℓf˚ pcq år0,1´ǫs q´1 Òf c | βc is close to c rel C.

rc by the intertwiner FE,D (see (9)), it suffices
Proof. Since Bc is obtained from B
to consider the case E “ D “ H; and in that case, by Lemma 2.17 it suffices to
note that Bc is indeed an Hc -Gf˚ pcq -biset.

rÑA
r be an abstract portrait extending
Let H BG be an orbisphere biset; let f˚ : C
r
B˚ ; let deg : C Ñ N be an extension of degB : C Ñ N; and let pBc qcPCr be a portrait
of bisets in B with portraits of groups pHc qcPCr in H and pGa qaPAr in G. A congruence
of portraits is defined by a choice of phc qcPCr in H and pga qaPAr in G, and modifies
the portrait of bisets by replacing
Hc ù h´1
c Hc h c ,

Bc ù h´1
c Bc gf˚ pcq ,

Ga ù ga´1 Ga ga .

By Lemma 2.17, any two minimal portraits of bisets are congruent.
2.6. Main result. Consider an orbisphere map f : pS 2 , C, ordq Ñ pS 2 , A, ordq. We
r Ą
rĄ
call it compatible with FE : pS 2 , C,
ordq 99K pS 2 , C, ordq and FD : pS 2 , A,
ordq 99K
r Ñ A
r and deg : C
r Ñ N if degpeq “ 1 for all e P E with
pS 2 , A, ordq and f˚ : C
Ą degpcq | Ą
r and tdegpf @f ´1 paqqu “
f˚ peq P D, and ordpcq
ordpf˚ pcqq for all c P C,
´1
r Ą
ordq Ñ
tdegpf˚ paqu for all a P A. Equivalently, there is an orbisphere map fr: pS 2 , C,
2 r Ą
2
2
pS , A, ordq that can be isotoped within maps pS , C, ordq Ñ pS , A, ordq to a map
r Ñ N and f˚ : C
r Ñ N are induced by fr.
making (7) commute and such that deg : C
r Ñ H, FD : G
r Ñ G,
Compatibility of an orbisphere biset H BG with FE : H
r Ñ A,
r and deg : C
r Ñ N is defined similarly, and is equivalent to the existence
f˚ : C
r making (8) commute.
of a biset B
We are now ready to relate the mapping class bisets M ˚ to portraits of bisets:
r Ñ H and FD : G
r Ñ G be forgetful morphisms of
Theorem 2.19. Let FE : H
orbisphere groups as in (8), and let H BG be an orbisphere biset compatible with
r Ñ A,
r and deg : C
r Ñ N.
FE , FD , f ˚ : C

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

19

Then for every portrait of bisets pBc qcPCr in B parameterized by f˚ and deg there
r r mapping to B under FE,D with a minimal portrait
exists an orbisphere biset H
Ă BG
mapping to pBc qcPCr .
r is unique up to pre- and post-composition with bisets of knitting
Furthermore, B
mapping classes and we have a congruence of bisets (see (19))
(31)

π1˚ pY,Eq M

˚

pB|E, Dqπ1 pX,Dq
P

˚

H E ¸E Ó.

pB 1 P M pBq, pBc1 qcPCr q | B 1 P M pBq, pBc1 qcPCr a portrait in B 1

r 1 q “ pFE,D pB
r 1 q, induced portrait of B
r 1 q.
given by PpB

(

GD

The H E -GD -action on tpB 1 , pBc1 qcPCr qu is given by

phc qcPE pB 1 , pBc1 qcPCr qpgd qdPD “ pB 1 , phc Bc1 gf˚ pcq qcPCr q,

with the understanding that hc “ 1 if c R E and gf˚ pcq “ 1 if f˚ pcq R D, and the
˚
action of E Ó. is given by permutation of the portrait of bisets.
r “ G),
r Theorem 2.19 proves
In the dynamical situation (i.e. when H “ G and H
Theorem C.
˚

Proof of Theorem 2.19. Clearly, (31) is an intertwiner: firstly, the actions of E Ó.
are compatible; we may ignore them in the sequel. Secondly, let pBc1 qcPCr be the
r 1 . For e P E we may write
induced portrait of bisets of B
Be1 “ tme #p´1 | p : r0, 1s Ñ Y, p´1 p0q “ e, f ˝ p “ ℓf˚ peq u,

see (30). Consider ψ P ModpY |Eq; the action of ψ on Be1 replaces me by ψ˝me ; and
if EE pψq “ phe qePE then ψ ˝ me “ he me by the very definition of ‘push’ and (19).
The same argument applies to the right action.
Let us now show that P is a congruence. Since P is an intertwiner between leftfree bisets with isomorphic acting groups, it is sufficient to show that P preserves
left orbits.
We proceed by adding new points to D and E. If E “ H, then the forgetful
maps M ˚ pB|E, Dq Ñ M pBq and
(
pB 1 P M pBq, pBc1 qcPCr q | B 1 P M pBq, pBc1 qcPCr a portrait in B 1 Ñ M pBq

are bijections and the claim follows. Therefore, it is sufficient to assume that
D “ H.
By [5, Theorem 7.9], the biset B may be written as Bpf q for a branched covering
f : pS 2 , C, ordq Ñ pS 2 , A, ordq, unique up to isotopy rel C. We lift f to a branched
r ` be its biset and let pB
rc` qcPf ´1 pAq be
covering f ` : pS 2 , f ´1 pAqq Ñ pS 2 , Aq. Let B
its minimal portrait of bisets, which is unique by Lemma 2.17.
Let us show that for every portrait of bisets pBc qcPCr in B there is an orbisphere
r whose minimal portrait of bisets maps to pBc q. This B
r will be of the form
biset B
`
2
´1
r
Bpmq b Ff ´1 pAqzC,H
p
B
q
for
a
homeomorphism
m
:
pS
,
f
pAqq
ý.
r
rc` in t¨u b Ă B
r ` – t¨u bH B, and compare
First, consider the images of all B
H
r
them to the images of all Bc . The condition that as c1 ranges over f˚´1 pf˚ pcqq the

20

LAURENT BARTHOLDI AND DZMITRY DUDKO

Bc1 lie in different H-orbits of B lets us select which preimages of A correspond
r and thus produces a well-defined map C
r Ñ f ´1 pAq.
to the marked points in C,
r to f ´1 pAq in the specified manner;
Let m1 be an isotopy of pS 2 , Cq which maps C
1 `
2 r
2
r 0 and portrait of bisets pB
r0 q r ;
then m f : pS , C, ordq Ñ pS , A, ordq has biset B
c cPC
0
0
r q Ď HBc for all c P C,
r so we may write hc FE,H pB
r q “ Bc for
and FE,H pB
c
c
some hc P H. (We recall that Bc consists of dpcq elements permuted by Gf˚ pcq ,
r 0 q “ Bc if and only if
where dpcq is the local degree of f at c. We have hc FE,H pB
c
ś
0
0
r
r q X Bc ‰ H.) We set B
r“
pushph
q
B
,
and
note that the minimal
hc FE,H pB
c
c
cPE
r
portrait of bisets of B maps to pBc qcPCr under FE,H .
The only choice involved is that of a mapping class that yields pushphc q when
restricted to C Y tcu, namely that of knitting mapping classes.


2.7. Fibre bisets. Consider an orbisphere map fr: pS 2 , C \ E, Ą
ordq Ñ pS 2 , A \
Ą
D, ordq as above and define the saturation of E as
ğ
˘
`
s :“
E
fr´1 frpeq zC.
ePE

Ą be an orbisphere map and
Lemma 2.20. Let fr: pS 2 , C \E, Ą
ordq Ñ pS 2 , A\D, ordq
2
s
s
let m : pS , C \ Eq ý be a homeomorphism such that m åC “ 1 and for every e P E
˚
2
r
r
we have f pmpeqq “ f peq. If the isotopy class of m is not in Mod pS , C \ Eq,
s
namely if m moves at least one point in E to EzE,
then mfr ﬀC\E fr.
˚ r
s ý as above
For every r
g P M pf q there is a homeomorphism m : pS 2 , C \ Eq
r
r
r
s
(i.e. m åC “ 1 and f pmpeqq “ f peq for e P E) such that r
g «C\E mf .

s does not act on M ˚ pfrq: there are orbisphere maps
Note that Mod˚ pS 2 , C \ Eq
2
fr1 , fr2 : pS , C \ E, Ą
ordq Ñ pS 2 , A \ D, Ą
ordq such that fr1 «C\E fr2 but mfr1 ﬀC\E
2
r
s ý as above.
mf2 for a homeomorphism m : pS , C \ Eq

s is saturated, by Lemma 2.9(2) there is an
Proof. Suppose mfr «C\E fr. Since E
2
s
s
n : pS , C \ Eq ý with n åC\E “ 1 and frpnpeqq “ frpeq for e P EzE
such that
r
r
nmf «C\Es f . This contradicts Lemma 2.9(1): the biset
s Ñ pS 2 , A \ Dqq
(32)
M ˚ pfr: pS 2 , C \ Eq

is left-free while nm ‰ 1.
The second claim follows from Lemma 2.9(2) applied to the biset (32).
We are interested in the fibre biset

F ´1 pB 1 qGD
H E E,D



under the forgetful map
r
FE,D : M pB|E, Dq Ñ M pBq. For every a P A define Ea :“ E X f˚´1 paq, where
rÑA
r is the portrait.
f˚ : C
˚

Proposition 2.21. We have
ź
ź
´1
˚
1
˚
1
(33) H E FE,D
pB 1 qGD –
H Ea M pB |Ea , Hq1 ˆ
H Ed M pB |Ed , tduqGtdu .
aPA

1

dPD

Suppose that B is the biset of g : pS , C, ordq Ñ pS 2 , A, ordq; then
The biset

2

˚
1
Ea
ˆ tι : Ea ãÑ g ´1 paqzCu.
H Ea M pB |Ea , Hq1 – H
˚
1
H Ed M pB |Ed , tduqGtdu is congruent to the biset
tpbe qePEd P B 1Ed | Hbe ‰ Hbe1 if e ‰ e1 u

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

21

endowed with the actions
phe qePEd ¨ pbe qePEd ¨ g “ phe be gqePEd .
´1
pB 1 qGD is isomorphic to the biset of
Proof. By Theorem 2.19, the biset H E FE,D
1
portraits pBc qcPCr in B . Recall that pBc qcPC is fixed (by Lemma 2.17) while pBe qePE
varies; this shows (33).
The second claim follows from Lemma 2.20.
For d P D and e P Ed we have Be “ tbe u; i.e. the choice of pBe qePEd is equivalent
to the choice of pbe qePEd P B 1Ed subject to the condition stated above.


Note that H Ea M ˚ pB 1 |Ea , Hq1 will not be transitive, as soon as there are at least
two maps ι : Ea Ñ g ´1 paqzC.
Let us define
r b ModpX|Dq.
M pB|E, Dq :“ ModpY |Eq b M pBq

Theorem 2.22. Suppose f˚ pEq Ă D and, moreover, that f˚ : E Ñ D is a bijection.
We then have a congruence
` ´1
˘
(34)
FE,D pB 1 q GD Ñ pH B 1 G qE
HE

rc q r in B 1 to pbe qePE where B
re “ tbe u. The group GD is
mapping the portrait pB
ePC
identified with GE via the bijection f˚ : E Ñ D.
r “ M pBq,
r M ˚ pB|E, Dq “ M pB|E, Dq, and exact sequences
Moreover, M ˚ pBq
similar to (27) hold:
(35)
r
M pBq
knBraidpY,Eq

knBraidpX,Dq

r

BraidpY,Eq M pBqBraidpX,Dq
EE,D

r
M pBq

π1 pY,Eq M pB|E, Dqπ1 pX,Dq

P

M pB|E, Dq

FE,D
FE,D

HE

M pBq

!

–

pB 1 P M pBq, pBc1 qcPCr q
pBc1 qcPCr is a portrait in B 1

)

The bottom sequence in (35) can be written (using (34)) as
(36)

Ů

E

B 1 PB

pB 1 q

M pB|E, Dq

M pBq.

Proof. The first claim follows from Proposition 2.21. Since pB 1 qE is a transitive
biset, we obtain M pB|E, Dq “ M ˚ pB|E, Dq and the exact sequences (36) and (35)
hold because the fibres are connected.


.
GD

22

LAURENT BARTHOLDI AND DZMITRY DUDKO

Corollary 2.23. Let f : pS 2 , A, ordq ý be an orbisphere map and let D “
be a finite union of periodic cycles Di of f . Then (36) takes the form
ğ ğ
(37)
Bpgqbp#Di q ãÑ M pf |D, Dq ։ M pf q. 

Ů

iPI

Di

gPMpf q iPI

2.8. Centralizers of portraits. Let us now consider the dynamical situation:
r “ H;
r we abbreviate FD “ FD,D and M pB|Dq “ M pB|D, Dq.
H “ G and G
r we denote by ZpBq
r ď ModpGq
r the centralizer of
Given an orbisphere biset B
r
r
r
r in
B in M pBq and by ZpB|Dq ď ModpG|Dq the centralizer of the image of B
M pB|Dq, see Theorem 2.19. We have a natural forgetful map
r Ñ ZpB|Dq
r
ZpBq

(38)

which is, in general, neither injective nor surjective. However, we will show in
r is geometric non-invertible.
Corollary 4.28 that (38) is an isomorphism if B
Recall from (12) the short exact sequence π1 pX, Dq ãÑ ModpG|Dq ։ ModpGq.
We have the corresponding sequence for centralizers:
(39)

1

r
ZpB|Dq
X π1 pX, Dq

r
ZpB|Dq

ZpBq.

r is geometric non-invertible, then ZpB|Dq
r
r
If B
X π1 pX, Dq is trivial, so ZpB|Dq
–
r is naturally a subgroup of ZpBq, and in Theorem 4.41 we will show that it
ZpBq
has finite index.

Definition 2.24. The relative centralizer ZD ppBa qaPAr q of a portrait of bisets
pBa qaPAr is the set of pgd q P GD such that
Bd “ gd´1 Bd gf˚ pdq for all d P D,

with the understanding that gf˚ pdq “ 1 if f˚ pdq R D.

△

We remark that we could also have defined the “full” normalizer, consisting of
r
r and its subgroup
all pga q P GA with Ggaa “ Ga and Ba “ ga´1 Ba gf˚ paq for all a P A,
´1
the “full” centralizer, in which ga centralizes Ga and ga ¨ ´ ¨ gf˚ paq is the identity
on Ba ; but ś
we will make no use of these notions. The “full” normalizer is the direct
product of aPA Ga and the relative centralizer.
We also note that, if pgd qdPD belongs to the relative centralizer of pBa qaPAr and
f n pdq P A for some n P N, then gd “ 1.
Applying Theorem 2.19 to the dynamical situation we obtain:
r r ։ G BG be a forgetful morphism and let pBa q r be
Proposition 2.25. Let Gr B
G
aPA
the induced portrait of bisets in B. Then any choice of isomorphisms π1 pX, dq – G
gives an isomorphism
–

r
ZpB|Dq
X π1 pX, Dq ÝÑ ZD ppBa qaPAr q.



3. G-spaces

We start by general considerations. Let Y be a right H-space, and let X be a
right G-space. For every map f : Y Ñ X there exists a natural H-G-biset M pf q,
defined by
(40)

M pf q :“ thf g :“ f p´ ¨ hq ¨ g | h P H, g P Gu,

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

23

namely the set of maps Y Ñ X obtained by precomposing with the H-action and
post-composing with the G-action. Note that M pf q is right-free if the action of G
is free on X. We have a natural G-equivariant map Y bH M pf q Ñ X given by
evaluation: y b b ÞÑ bpyq. Define
Hf :“ th P H | Dg P G with hf “ f g in Mf u

the stabilizer in H of f G P M pf q{G. Then f descends to a continuous map
f : Y {Hf Ñ X{G.
Lemma 3.1. Suppose that X and Y are simply connected and that G, H act freely
with discrete orbits. Then M pf q is isomorphic to the biset of the correspondence
f

Y {H Ð Y {Hf Ñ X{G as defined in [3].

Proof. Let us define the following subbiset of M pf q:

(41)

M 1 pf q :“ tf p´ ¨ hq ¨ g | h P Hf , g P Gu.

Since Y, X are simply connected, the biset of f : Y {Hf Ñ X{G is isomorphic to
M 1 pf q. The isomorphism is explicit: choose basepoints : P Y and ˚ P X so that
π1 pY {H, :Hq – H and π1 pX{G, ˚Gq – G. Given b P Bpf q, represent it as a
path β : r0, 1s Ñ X{G with βp0q “ f p:Hf q and βp1q “ ˚G, and lift it to a path
β : r0, 1s Ñ X. We have βp0q “ f p:qh and βp1q “ ˚g for some h P Hf , g P G, and
we map b P Bpf q to h´1 ¨ f ¨ g P M 1 pf q. This map is a bijection because both Bpf q
and M 1 pf q are right-free. We finally have
M pf q – H bHf M 1 pf q – BpY {H Ð Y {Hf Ñ X{Gq.



In case the actions of G, H are discrete but not free, there still is a surjective
morphism BpX{G Ø Y {Hq ։ M pf q, when X{G and Y {H are treated as orbispaces.
Example 3.2 (Modular correspondence). The mapping class biset M pf q from §2.4
is isomorphic to the biset of the associated correspondence on Moduli space, see [5,
Proposition 8.1]. Indeed, M pf q is identified with
(42)

tσn ˝ σf ˝ σm | n P ModpS 2 , Cq, m P ModpS 2 , Aqu,

where σm : TA ý, σn : TC ý, and σf : TA Ñ TC are the pulled-back actions
between Teichmüller spaces of m, n, f respectively. By Lemma 3.1, the biset (42)
is isomorphic to the biset of the modular correspondence
σf

i

MC ÐÝ Wf ÝÑ MA .
3.1. Universal covers. Let us now generalize Lemma 3.1 by dropping the requirement that X, Y be simply connected. Choose basepoints :, ˚, write Q “ π1 pY, :q
r “ π1 pY {H, :Hq and G
r “ π1 pX{G, ˚Gq; so we have exact
and P “ π1 pX, ˚q and H
sequences
(43)

π
rÑ
1ÑQÑH
H Ñ 1,

π
rÑ
1ÑP ÑG
G Ñ 1.

The universal cover of X may be defined as

r :“ tβ : r0, 1s Ñ X | βp1q “ ˚u{«;
X

it has a natural basepoint r̊ given by the constant path at ˚, and admits a right
r Ñ X is a covering,
P -action by right-concatenation of loops at ˚. The projection X

24

LAURENT BARTHOLDI AND DZMITRY DUDKO

r _ the left P -set structure on X,
r with
and is given by β ÞÑ βp0q. We denote by X
´1 _
_
´1
_ :“
r with tβ | β P Xu
r and
pβ ¨ g q . We may naturally identify X
action g ¨ β
its natural left P -action.
r Yr of X, Y respectively and a lift fr of f .
Let us consider the universal covers X,
We have the following situation, with the acting groups represented on the right,
and omitted in the left column:
Yr

fr

Yr {Q – Y

f

r ìG
r
X
XìG

Y {Hf

f

r – Y {H
Yr {H

X{G

r f be the full preimage of Hf in H.
r Note that H
r f is the stabilizer of frG
r in
Let H
r
r
r
r
r
r
r
M pf q{G: we have hf P f G if and only if πphqf P f G by the unique lifting property.
By Lemma 3.1, we have
r Ð Yr {H
r f Ñ X{
r Gq
r – BpY {H Ð Y {Hf Ñ X{Gq,
M pfrq – BpYr {H

and we shall see that M pfrq is an extension of bisets. We have a natural map
r g P G.
r
π : M pfrq Ñ M pf q, given by hfrg ÞÑ πphqf πpgq for all h P H,

Lemma 3.3. There is a short exact sequence of bisets
(44)

r

Q M pf qP

M pfrq

π

M pf q,

in which every fibre π ´1 phf gq is isomorphic to a twisted form Bphf gq of the biset
of f .
r with actions of
Proof. This follows immediately from Lemma 3.1 applied to Yr , X
Q, P respectively.


Let us now assume that the short exact sequences (43) are split, so π1 pX{Gq –
P ¸ G and π1 pX{Hq – Q ¸ H. We shall see that the sequence (44) is split, and
that some additional structure on M pf q and Bpf q allow the extension M pf q –
BpY {H Ø X{Gq to be reconstructed as a crossed product.
The splitting of the map π : π1 pX{G, ˚Gq Ñ G means that there is a family
tαg ugPG of curves in X such that αg connects ˚g to ˚ and αg1 g2 « pαg1 g2 q#αg2 .
Similarly there is a family tβh uhPH of curves in Y such that βh connects :h to :
and βh1 h2 « pβh1 h2 q#βh2 .
For every h P Hf there is a unique element of G, written hf P G, such that
h ¨ f “ f ¨ hf in M 1 pf q. For every h P Hf and every b P Bpf q, represented as a path
b : r0, 1s Ñ X with bp0q “ f p:q and bp1q “ ˚, define
bh :“ pf ˝ βh´1 q#pb ¨ hf q#αhf .
f

We clearly have pq ¨ b ¨ pqh “ q h ¨ bh ¨ ph , so Hf acts on Bpf q by congruences. We
´1
convert that right action to a left action by h b :“ bh .

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

25

For every c P M 1 pf q and every p P P , write c “ f g and define c p :“ g p “ gpg ´1 .
We clearly have cg p “ c pg pq.
Lemma 3.4. The biset of f : Y {Hf Ñ X{G is isomorphic to the crossed product
r f – Q ¸ Hf and
Bpf q ¸ M 1 pf q, which is Bpf q ˆ M 1 pf q as a set, with actions of H
r – P ¸ G given by
G
As a consequence,

pq, hq ¨ pb, cq ¨ pp, gq “ pq ¨ h pb ¨ c pq, h ¨ c ¨ gq.

f
r b Ă Bpf q ¸ M 1 pf q.
BpY {H Ð Y {Hf Ñ X{Gq – H
Hf

Proof. This is almost a tautology. The short exact sequence (44) splits, with section
r H
r on X,
r Yr can be identified with
h ¨ f ¨ g ÞÑ p1, hq ¨ fr ¨ p1, gq, and the actions of G,
concatenation of lifts of the paths αg , βh .


3.2. Self-similarity of G-spaces. We return in more detail the situation in which
r so as to keep Y, X for the original
Y, X are universal covers; we rename them to Yr , X
space.
Consider two pointed spaces pY, :q and pX, ˚q with H :“ π1 pY, :q and G :“
π1 pX, ˚q, and let f : Y Ñ X, be a continuous map. Recall that its biset is defined
by
(45)

H Bpf qG

:“ tβ : r0, 1s Ñ X | βp0q “ f p:q, βp1q “ ˚u{«,

with the natural actions by left- and right-concatenation. We thus naturally have
r with corresponding right actions, and left action given by composing
Bpf q Ď X,
loops via f .
The map f : Y Ñ X naturally lifts to a map Yr Ñ Y Ñ X, and every choice
of β P Bpf q defines uniquely, by the lifting property of coverings, a lift frβ : Yr Ñ
r with the property that r: ÞÑ β. Furthermore, we have the natural identities
X
r
fβ p´¨hq¨g “ frhβg , so that the biset Bpf q as defined in (45) is canonically isomorphic
to every biset M pfrβ q as defined in (40), when an arbitrary β P Bpf q is chosen, and
r of f is chosen.
to Bpfrq, when an arbitrary lift fr: Yr Ñ X
r choosing fr “ 1
If f : Y Ñ X is a covering, then we may assume Yr “ X;
r
gives the simple description Bpf q “ H GG . Recall that the biset of a correspondence pf, iq : Y Ð Z Ñ X is defined as Bpf, iq “ Bpiq_ b Bpf q. In the case
of a covering correspondence, in which f is a covering, we therefore arrive at
Bpf, iq “ Bpiq_ bπ1 pZq G. We shall give now a more explicit description of this
biset using covering spaces, just as we had in [3, (15)]
Bpf, iq “ tpδ : r0, 1s Ñ Y, p P Zq | δp0q “ :, δp1q “ ippq, f ppq “ ˚u{«

and the special case, when i : Z Ñ Y is injective, of

Bpf, iq “ tδ : r0, 1s Ñ Y | δp0q “ :, f pi´1 pδp1qqq “ ˚u{«.

Still assuming that f : Z Ñ X is a covering map, define
rH :“ tpδ, pq P Yr ˆ Z | ippq “ δp0qu,
(46)
Z

the fibred product of Yr with Z above Y , see Diagram (48) left. (Note that ZrH is
not the universal cover of Z.) In case i : Z Ñ Y is injective, we may write
rH “ tδ P Yr | δp0q P ipZqu,
Z

26

LAURENT BARTHOLDI AND DZMITRY DUDKO

rH is the full preimage of ipZq under the covering map Yr Ñ Y .
so Z

Proposition 3.5. If pf, iq is a covering correspondence, the following map defines
an isomorphism of H-spaces:
Φ:

H Bpf, iqG

r _ Ñ pZrH q_ given by Φppδ, pq b α_ q “ ppδ#pi ˝ α´1 Òp qq´1 , pq.
bX
f

For every b “ pδ, pq P Bpf, iq the map

r _ Ñ pZ
rH q_ given by α_ ÞÑ Φpb b α_ q
frb´1 : X

is the unique lift of the inverse of the correspondence Y Ð Z Ñ X mapping r̊ to b;
we have the equivariance properties
´1
frhbg
“ h ¨ frb´1 pg ¨ ´q.

(47)

The inverses and contragredients may seem unnatural in the statements above;
but are necessary for the actions to be on the right sides, and are justified by the
fact that we construct a lift of the inverse of the correspondence, rather than the
correspondence itself:

(48)

Yr _

ri

i

Y

pZrH q_

frb´1

Z

f

r_
X
X

rH ,
Proof. It is obvious that Φ is H-equivariant, and it is surjective: given pδ, pq P Z
p
p
´1
´1
´1
r with f ppq “ αp0q, and write δ “ pα Ò q #α Ò #δ,
choose a path α P X
f
f
expressing in this manner pδ, pq_ “ Φpppα´1 Òpf #δq´1 , α´1 Òpf p1qq b α_ q.
If Φppδ, pq b α_ q “ Φppδ 1 , p1 q b pα1 q_ q, then α and α1 start at the same point,
so α “ α1 g for some g P G, and we have then pδ, pq “ pδ 1 , p1 qg ´1 so pδ, pq b α_ “
pδ 1 , p1 q b pα1 q_ and Φ is injective.
It is easy to see that frb´1 is a lift of f ´1 . Conversely, every lift fr´1 of f ´1
maps r̊ to an element b P Bpf, iq because fr´1 pr̊_ q ends at an f -preimage of ˚ by
construction; and then fr´1 “ frb´1 by unicity of lifts. Finally, equivariance follows
´1
from frb´1 pgα_ q “ Φpb b gα_ q “ Φpbg b α_ q “ frbg
pα_ q.


r ì G is a
3.3. Planar discontinuous groups. A planar discontinuous group X
r for
group acting properly discontinuously on a plane, which will be denoted by X:
every bounded set V the set tg P G | V g X V ‰ Hu is finite.
Let X :“ pS 2 , A, ordq be an orbifold with non-negative Euler characteristic,
consider ˚ P S 2 zA a base-point, and write G :“ π1 pX, ˚q. Then the universal cover
rG of X is a plane endowed with a properly discontinuous action of G. We denote
X
r Ñ X the covering map.
by π : X
By the classification of surfaces, there are only two cases to consider: X “ C
with G a lattice in the affine group tz ÞÑ az ` b | a, b P C, |a| “ 1u, and X “ H the
upper half plane, with G a lattice in PSL2 pRq.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

27

Consider another orbisphere Y :“ pS 2 , C, ordq and a branched covering f : Y Ñ
X, and fix basepoints : P Y and ˚ P X with corresponding fundamental groups
H “ π1 pY, :q and G “ π1 pX, ˚q. Let H Bpf qG be the biset of f . As usual, we
view f as a correspondence Y Ð Z Ñ X with Z “ pS 2 , f ´1 pAq, ordq and i a
homeomorphism S 2 Ñ S 2 mapping a subset of f ´1 pAq to C. The fibred product
rH constructed in (46) is naturally a subset of the plane Yr , with orbispace points
Z
and punctures at all H-orbits of f ´1 pAqzC. We need the following classical result.
Theorem 3.6 (Baer [26, Theorem 5.14.1]). Every orientation-preserving homeomorphism of a plane commuting with a properly discontinuous group action is
isotopic to the identity along an isotopy commuting with the action.

Let us reprove [5, Theorem 7.9] using our language of group actions:
Corollary 3.7. Let two orbisphere maps f, g : pS 2 , C, ordq Ñ pS 2 , A, ordq have
isomorphic orbisphere bisets. Then f «C g.
In other words, the orbisphere biset of f : pS 2 , C, ordq Ñ pS 2 , A, ordq is a complete invariant of f up to isotopy rel C.
Proof. Let us write X “ pS 2 , A, ordq and Y “ pS 2 , C, ordq and G “ π1 pX, ˚q
and Y “ π1 pY, :q. We may represent f, g respectively by covering pairs pf, iq
and pg, iq, with coverings f, g : pS 2 , P, ordq Ñ pS 2 , A, ordq and i : pS 2 , P, ordq Ñ
rH its fibred product
pS 2 , C, ordq. Let us furthermore write Z “ pS 2 , P, ordq and Z
with Yr . Identifying Bpf, iq and Bpg, iq, choose b P Bpf, iq “ Bpg, iq, and let
r _ Ñ pZ
rH q_
frb´1 , r
gb´1 : X

be the corresponding lifts as in Proposition 3.5.
Since i is injective, the map pZrH q_ Ñ Z is a covering, so frb´1 and grb´1 are
coverings. We may therefore consider their quotient frb ˝ grb´1 , which is a wellr _ ý, and is normalized to preserve the base point r̊. By (47) it is
defined map pXq
a homeomorphism commuting with the action of G.
By Theorem 3.6 there is an isotopy pr
h´1
b,t qtPr0,1s of maps satisfying (47) between
´1
r
1 and fb ˝ grb .
Since the set of fixed points of G is discrete, r
h´1 preserves all fixed points of G
b,t

and therefore projects to an isotopy pht qtPr0,1s in X. We have h0 “ 1 and h1 ˝g “ f ,
so the maps f and g are isotopic rel B.

4. Geometric maps

Let M be a matrix with integer entries and with detpM q ą 1. We call M
exceptional if one of the eigenvalues of M lies in p´1, 1q; so the dynamical system
M : R2 ý has one attracting and one repelling direction.
For r P R2 , denote by xZ2 , ´z ` ry the group of affine transformations of R2
generated by translations z ÞÑ z ` v with v P Z2 and the involution z ÞÑ ´z ` r.
The quotient R2 {xZ2 , ´z `ry is a topological sphere, with cone singularities of angle
π at the four images of 21 pr ` Z2 q.
We call a branched covering f : S 2 ý a geometric exceptional map if f : S 2 ý
is conjugate to a quotient of an exceptional linear map M : R2 ý under the action
of xZ2 , ´z ` ry for some r P R2 ; in particular, p1 ´ M qr P Z2 . A Thurston map
f : pS 2 , A, ordq ý is called geometric if the underlying branched covering f : S 2 ý

28

LAURENT BARTHOLDI AND DZMITRY DUDKO

is either Böttcher expanding (see [4, Definition 4.1]; there exists a metric on S 2 that
is expanded everywhere except at critical periodic cycles), a geometric exceptional
map, or a pseudo-Anosov homeomorphism. We refer to the first two types as noninvertible geometric maps.
We may consider more generally affine maps z ÞÑ M z ` q on R2 , and then their
quotient by the group xZ2 , ´zy; the map z ÞÑ M z on R2 {xZ2 , ´z ` ry is converted
to that form by setting q “ pM ´ 1qr. Conversely, if 1 is not an eigenvalue of M ,
then we can always convert an affine map into a linear one, at the cost of replacing
the reflection ´z by ´z ` r in the acting group.
Lemma 4.1. Let M : R2 ý be exceptional. Then for every bounded set D Ă R2
containing p0, 0q there is an n ą 0 such that for every m ě n we have M ´m pDq X
Z2 “ tp0, 0qu. Moreover, n “ npDq can be taken with npDq ď log diampDq.
Proof. Let λ1 , λ2 be the eigenvalues of M , and let e1 and e2 be unit-normed eigenvectors associated with λ1 and λ2 . It is sufficient to assume that D is a parallelogram centered at p0, 0q with sides parallel to e1 and e2 :
(49)

D “ tv P R2 | v “ t1 e1 ` t2 v2 with |t1 | ď x and |t2 | ď yu.

In particular, the area of D is comparable to xy. Then M ´n pDq is again a parallelogram centered at p0, 0q with sides parallel to e1 and e2 .
We claim that there is δ ą 0 such that if D is a parallelogram of the form (49)
with areapDq ă δ, then D X Z2 “ tp0, 0qu. This will prove the lemma because
areapM ´n pDqq “ areapDq{pdet M qn and log diampDq — log areapDq.
Without loss of generality we assume x ą y, so D is close to Re1 . Let µ1 be the
slope of Re1 . Since M is exceptional, the numbers λ1 , λ2 , µ1 are quadratic irrational,
so they are not well approximated by rational numbers: there is a positive constant
C such that |µ1 ´ pq | ą qC2 for all pq P Q.
Suppose that D contains a non-zero integer point w “ pp, qq; so x ě |q|. Also,
w is close to Re1 , and in particular q ‰ 0 if δ is sufficiently small; we may suppose
q ą 0. It also follows that µ1 is close to pq . The distance from w to Re1 is
dpw, Re1 q ď y —
On the other hand,

Combining, we get

areapDq
.
x

ˇ
p ˇˇ
ˇ
dpw, Re1 q “ |w| sin ?pw, e1 q ě q ˇµ1 ´ ˇ.
q

ˇ
areapDq
δ
p ˇˇ dpw, Re1 q
ˇ
ď
ď 2,
ˇµ1 ´ ˇ ď
q
q
q¨x
q
and the claim follows for δ ăă C.
2


2

Corollary 4.2. Let f : pS , Aq ý be a geometric exceptional map, let p P S zA be
a periodic point with period n, and let γ P π1 pS 2 zA, pq be a loop starting and ending
at p. Let |γ| be the length of γ with respect to the Euclidean metric of the minimal
orbifold structure pPf , ordf q. If γ is trivial rel pPf , ordf q and m ą log |γ|, then the
lift γÒpf nm is a trivial loop rel A.
Proof. By passing to an iterate, we may assume that p is a fixed point of f . Since
f is geometric exceptional, we have a branched covering map π : R2 Ñ S 2 under

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

29

which f lifts to an exceptional linear map M , and we may assume πp0, 0q “ p. By
q
assumption, γÒp0,0q
is a trivial loop in R2 . By Lemma 4.1, the sets M ´m pγÒp0,0q
π
π
´1
and π pAq are disjoint for m ą log |γ|; hence γÒpf nm is a trivial loop rel A.

Geometric exceptional maps all admit a minimal orbifold modeled on R2 {xZ2 , ´zy,
which has cone singularities of angle π at the images of t0, 12 u ˆ t0, 12 u. We consider
this class in a little more detail:
4.1. p2, 2, 2, 2q-maps. A branched covering f : pS 2 , Pf , ordf q ý is of type p2, 2, 2, 2q
if |Pf | “ 4 and ordf pxq “ 2 for every x P Pf . In this case, f is isotopic to a quotient
of an affine map z ÞÑ M z ` q under the action of xZ2 , ´zy, see Proposition 4.6(B).
Lemma 4.3. Let M be a matrix with integer entries and detpM q ą 1. Denote by
λ1 and λ2 the eigenvalues of M , ordered as |λ2 | ě |λ1 | ą 0. Then the following are
all mutually exclusive possibilities:
‚ if M is exceptional, that is 0 ă |λ1 | ă 1 ă |λ2 | and λ1 , λ2 P R, then
M : R2 ý is expanding in the direction of the eigenvector corresponding to
λ2 and M : R2 ý is contracting in the direction of the eigenvector corresponding to λ1 ;
‚ if λ1 P t˘1u, then M 2 : R2 ý preserves the rational line tz P R2 | M z “ zu;
‚ the map M : R2 ý is expanding in all other cases; that is λ1 “ λ2 R R, or
λ1 , λ2 P R but |λ1 |, |λ2 | ą 1.
Proof. If M ’s eigenvalues are non-real, then λ1 “ λ2 and |λ1 | “ |λ2 |, so M is
expanding. If λ1 and λ2 are real, then they are of the same sign and their product
is greater than 1. The lemma follows.

The following lemma follows from [23, Main Theorem II]:
Lemma 4.4. If f : pS 2 , Pf , ordf q ý is doubly covered by a torus endomorphism
z ÞÑ M z ` q, then f is geometric if and only if f is Levy-free.
Proof. We consider the exclusive cases of Lemma 4.3. In the first case, f is geometric and z ÞÑ M z ` q preserves transverse irrational laminations (given by the
eigenvectors of M ), so z ÞÑ M z ` q admits no Levy cycle and a fortiori neither
does f .
In the second case, f is not geometric and admits as Levy cycle the projection
of the 1-eigenspace of M .
In the third case, f is expanding, and admits no Levy cycle by [4, Theorem A].

We shall mainly study p2, 2, 2, 2q-maps algebraically: we write the torus as R2 {Z2
and the p2, 2, 2, 2q-orbisphere as R2 {xZ2 , ´zy. Its fundamental group G is identified
with xZ2 , ´zy – Z2 ¸ t˘1u. The orbifold points are the images on the orbisphere
of t0, 21 u ˆ t0, 12 u. We start with some basic properties of G:
Proposition 4.5.
(50)

(A) Every injective endomorphism of G is of the form

v

M : pn, 1q ÞÑ pM n, 1q and pn, ´1q ÞÑ pM n ` v, ´1q

for some v P Z2 and some non-degenerate matrix M with integer entries.
We have
N w ˝ M v “ pN M qw`N v .

30

LAURENT BARTHOLDI AND DZMITRY DUDKO

There are precisely 4 order-2 conjugacy classes in G, each of the form
pa, ´1qG “ tpa ` 2w, ´1q | w P Z2 u for some a P t0, 1u ˆ t0, 1u Ă Z2 .
The set of order-2 conjugacy classes of G is preserved by M v .
(B) The automorphism group of G is tM v | det M “ ˘1u and is naturally
identified with Z2 ¸ GL2 pZq. The inner automorphisms of G are identified
with 2Z2 ¸ t˘1u, and the outer automorphism group of G is identified
with pZ{2Zq2 ¸ PGL2 pZq. The index-2 subgroup of positively oriented outer
automorphisms is Out` pGq “ pZ{2Zq2 ¸ PSL2 pZq.
(C) The modular group ModpGq is free of rank 2, and we have
ModpGq “ tM v | detpM q “ 1, M ” 1 pmod 2q, v P 2Z2 u{p˘1q2Z

2

– tM 0 | M ” 1 pmod 2qu{t˘1u.

(D) Two bisets BM v and BN w are isomorphic if and only if M “ ˘N and
pM v q˚ “ pN w q˚ as maps on order-2 conjugacy classes, if and only if M “
˘N and v ” w pmod 2Z2 q.
We remark that (50) also follows from Lemma 3.4.
Proof. (A). It is easy to check that M v defines an injective endomorphism. Conversely, let M 1 : G Ñ G be an injective endomorphism. Then M 1 p0, ´1q “ pv, ´1q
for some v P Z2 because all pw, ´1q have order 2. On the other hand, M 1 åZ2 ˆt1u “
M åZ2 for a non-degenerate matrix M with integer entries because M 1 is injective.
It easily follows that M 1 “ M v as given by (50).
The claims on composition and order-2 conjugacy classes of G follow from direct
computation.
2
(B). Follows directly from (A) and the identification of G with t˘1uZ .
`
(C). We use (B); the modular group of G is the subgroup of Out pGq “ pZ{2q2 ¸
PSL2 pZq that fixes the order-2 conjugacy classes. The action of pZ{2Zq2 on these
classes is simply transitive, so the set of order-2 classes may be put in bijection with
pZ{2q2 under the correspondence pa ` 2Z2 , ´1q Ø a; then the action of PSL2 pZq
on order-2 conjugacy classes is identified with the natural linear action (noting
that ´1 acts trivially mod 2). It follows that ModpGq is the congruence subgroup
tM P PSL2 pZq | M ” 1 pmod ˘ 2qu, and it is classical that it is a free group of
rank 2.
(D). The bisets BM v and BN w are isomorphic if and only the maps M v , N w are
conjugate by an inner automorphism; so the claimed description follows from (B).

We turn to p2, 2, 2, 2q-maps, and their description in terms of the above; we use
throughout G “ Z2 ¸ t˘1u:
Proposition 4.6. Let f be a p2, 2, 2, 2q-map with biset
Then

G BG

“ Bpf, Pf , ordf q.

(A) The biset B is right principal, and for any choice of b0 P B there exists an
injective endomorphism M v of G satisfying gb0 “ b0 M v pgq for all g P G.
(B) The map f is Thurston equivalent to a quotient of z ÞÑ M z ` 21 v : R2 ý
under the action of G.
(C) The map f is Levy obstructed if and only if M has an eigenvalue in t˘1u.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

31

(D) If f is not Levy obstructed then for every b0 P B, writing M v as in (A)
and (50), the map f is Thurston equivalent to the quotient of z ÞÑ M z : Z2 ý
by the action of xZ2 , z ÞÑ ´z ` ry í R2 for a vector r P R2 satisfying
M r “ r ` v. The G-equivariant map of Proposition 4.17 takes the form

(51)

Φ:

G BG

b xZ2 ,´z`ry R2 Ñ xZ2 ,´z`ry R2 ,

b0 b z ÞÑ M ´1 z.

Proof. (A). Since f is a self-covering of orbifolds, G BG is right principal. The claim
then follows from Proposition 4.5(A).
(B). We claim that the quotient map, denoted by f¯, has a biset isomorphic to
G BG . Indeed, for g P G it suffices to verify that
pM z ` 21 vq ˝ gpzq “ M v pgq ˝ pM z ` 12 vq.

If g “ pt, 1q then both parts are M z ` M t ` 21 v, and if g “ p0, ´1q then both parts
are ´pM z ` 12 vq. Therefore, f and f¯ have isomorphic orbisphere bisets because
marked conjugacy classes are preserved automatically by Proposition 4.5 (A). By
Corollary 3.7 the maps f and f¯ are isotopic.
(C). Suppose that M has an eigenvalue in t1, ´1u; let w be the eigenvector of M
associated with this eigenvalue. Consider the foliation Fw of R2 parallel to w. Then
Fw is invariant under z ÞÑ M z ` 21 v as well as under the action of xZ2 , z ÞÑ ´zy.
Therefore, the quotient map has invariant foliation F w “ Fw {G. There are two
leaves in F w connecting points in pairs in the post-critical set of the quotient map;
any other leave is a Levy cycle.
(D). Since detpM ´ 1q ‰ 0, there is a unique r such that M r “ r ` v. It is easy
to see (as in (C)) that the quotient map in (E) has a biset isomorphic to G BG . By

Corollary 3.7 the quotient map is conjugate to f , and (51) is immediate.
Corollary 4.7. Let f be a p2, 2, 2, 2q-map. Then its biset Bpf : pS 2 , Pf , ordf q ýq
is of the form BM v for an endomorphism M v of G :“ Z2 ¸ t˘1u, namely it is G
qua right G-set, with left action given by g ¨ h “ M v pgqh.

Let us also recall how the biset of a p2, 2, 2, 2q-map is converted to the form
BM v . First, the fundamental group G is identified with Z2 ¸ t˘1u. The group
G has a unique subgroup H of index 2 that is isomorphic to Z2 , so H is easy to
find. The restriction of the biset to H yields a 2 ˆ 2-integer matrix M ; and the
translation part v is found by tracking the peripheral conjugacy classes. Note that
this procedure applies as well to non-invertible maps as to homeomorphisms; and
that the map is orientation-preserving precisely if detpM q ą 0.
4.2. Homotopy pseudo-orbits and shadowing. Let f : S 2 ý be a self-map,
and let I be an index set together with a index map also written f : I ý. An
I-poly-orbit pxi qiPI is a collection of points in S 2 such that f pxi q “ xf piq . If all
points xi are distinct, then pxi qiPI is an I-orbit.
Thus a poly-orbit differs from an orbit only in that it allows repetitions. Every
poly-orbit has a unique associated orbit, whose index set is obtained from I by
identifying i and j whenever xi “ xj . Note that we allow I “ N, I “ Z and
I “ Z{nZ as index sets, with f piq “ i ` 1, as well as I “ t0, . . . , m, . . . , m ` nu with
f piq “ i ` 1 except f pm ` nq “ m, an orbit with period n and preperiod m.
We shall consider a homotopical weakening of the notion of orbits. Our treatment
differs from [13] in a subtle manner (see below); recall that β «A,ord γ means that
the curves β, γ are homotopic in the orbispace pS 2 , A, ordq:

32

LAURENT BARTHOLDI AND DZMITRY DUDKO

Definition 4.8 (Homotopy pseudo-orbits). Let f : pS 2 , A, ordq ý be an orbisphere
self-map, and let I be a finite index set together with an index map also written
f : I ý.
An I-homotopy pseudo-orbit is a collection of paths
pβi qiPI with βi : r0, 1s Ñ S 2 zA satisfying βf piq p0q “ f pβi p1qq.

Two homotopy pseudo-orbits pβi qiPI and pβi1 qiPI are homotopic, written pβi qiPI «A,ord
pβi1 qiPI , if βi «A,ord βi1 for all i P I. In particular, βi p0q “ βi1 p0q and βi p1q “ βi1 p1q.
Two homotopy pseudo-orbits pβi qiPI and pγi qiPI are conjugate, written pβi qiPI „
pγi qiPI , if there is a collection of paths pℓi qiPI with
ℓi p0q “ βi p0q,

β p1q

ℓi p1q “ γi p0q and βi #ℓf piq Òf i

«A,ord ℓi #γi .

Poly-orbits are special cases of homotopy pseudo-orbits, in which the paths βi are
all constant.
△
Remark 4.9. A homotopy pseudo orbit can also be defined for an infinite I with
the assumption that pℓi qiPI and pβi qiPI in Definition 4.8 have uniformly bounded
length. Then Theorem 4.22 still holds.
In a homotopy pseudo-orbit, the curves pβi qiPI encode homotopically the difference between xi :“ βi p0q and a choice of preimage of xf piq . Note that Ishii-Smillie
define in [13, Definition 6.1] homotopy pseudo-orbits by paths βi connecting f pxi q
to xf piq ; their βi may be uniquely lifted to paths βi from xi to an f -preimage of
xf piq as in Definition 4.8, but our definition does not reduce to theirs, since our βi
are defined rel pA, ordq, not rel pf ´1 pAq, f ˚ pordqq.
βi

yi ¨

f

xf piq ¨

xi ¨
ℓi

βf piq

f

γi

f

¨ xf 2 piq

ℓf piq

yf piq ¨

γf piq

f

ℓf 2 piq
¨ yf 2 piq

Choose a length metric on S 2 , and define the distance between homotopy pseudoorbits by
`
˘
(52)
d pβi qiPI , pγi qiPI :“ max dpγi ptq, βi ptqq.
iPI,tPr0,1s

The following result states that the set of conjugacy classes of homotopy pseudoorbits is discrete:

Lemma 4.10 (Discreteness). Let pβi qiPI be a homotopy pseudo-orbit of f : pS 2 , A, ordq ý.
Then there is an ε ą 0 such every homotopy pseudo-orbit at distance less than ε
from pβi qiPI is conjugate to it.
Proof. Set ε “ mina‰bPA dpa, bq. Consider a homotopy pseudo-orbit pγi qiPI at distance δ ă ε from pβi qiPI . Connect βi p0q to γi p0q by a path ℓi of length at most δ.
Since SzA is locally contractible space, the curve ℓi is unique up to homotopy and
β p1q
βi #ℓf piq Òf i «A,ord ℓi #γi .

We recall that A8 denotes the union of all periodic cycles containing critical
points.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

33

Definition 4.11 (Shadowing). A homotopy pseudo-orbit pβi qPI shadows rel pA, ordq
a poly-orbit ppi qiPI in SzA8 if they are conjugate; namely if there are curves ℓi connecting βi p0q to pi that lie in S 2 zA except possibly their endpoints and such that
for every i P I we have
β p1q

ℓi «pA,ordq βi #ℓf piq Òf i

.

△

Lemma 4.12. The homotopy pseudo-orbit pβi qPI shadows the poly-orbit ppi qiPI if
and only if every neighbourhood pUi qiPI of ppi qiPI contains a homotopy pseudo-orbit
pβi1 qiPI conjugate to pβi qiPI ; namely, βi1 Ă Ui Q pi for all i P I.
Proof. If pβi q can be conjugated into a small enough neighbourhood of ppi qiPI , then
it is conjugate to ppi qiPI by Lemma 4.10. The converse is obvious.

Proposition 4.13. Suppose f : pS 2 , A, ordq ý is a geometric non-invertible map.
Then a periodic pseudo-orbit pβi qiPI shadows ppi qiPI rel pA, ordq if and only if pβi qiPI
shadows ppi qiPI rel pPf , ordf q.
Proof. Clearly if pβi qiPI shadows an orbit ppi qiPI rel pA, ordq then pβi qiPI shadows ppi qiPI rel pPf , ordf q. Conversely, suppose pβi qiPI shadows an orbit ppi qiPI rel
β p1q
pPf , ordf q, so there are paths ℓi connecting βi p0q to pi with ℓi «Pf ,ordf βi #ℓf piq Òf i .
β p1q

i
Thus ℓ´1
is a loop which is trivial rel pPf , ordf q, but may not be
i #βi #ℓf piq Òf
trivial rel A.
Consider the following pullback iteration. Set pβi0 qiPI :“ pβi qiPI and pℓ0i qiPI :“
0
pℓi qiPI . Define
`
`
βin´1 p1q ˘
βi p1q ˘
.
and pℓni qiPI :“ ℓn´1
Ò
pβin qiPI :“ βfn´1
f piq Òf
piq f
iPI
iPI

β p1q

Clearly the pβin qiPI are all conjugate. Observe that pℓni q´1 #βi #ℓnfpiq Òf i
passing through pi and

β p1q
pℓni q´1 #βi #ℓnfpiq Òf i

is a degree-1 preimage of
β p1q

If f is expanding, then the diameter of pℓni q´1 #βi #ℓnfpiq Òf i

is a loop

βi p1q
pℓn´1
q´1 #βi #ℓn´1
.
i
f piq Òf

tends to 0 expo-

β p1q
pℓni q´1 #βi #ℓnfpiq Òf i

nentially fast, hence
is trivial rel pA, ordq for all sufficiently
large n, and the claim follows.
β p1q
If f is exceptional, then pℓni q´1 #βi #ℓnfpiq Òf i is trivial rel pA, ordq for all sufficiently large n by Corollary 4.2.

At one extreme, homotopy pseudo-orbits include poly-orbits, represented as constant paths. At the other extreme, homotopy pseudo-orbits may be assumed to
consist of paths all starting at the basepoint ˚. As such, these paths represent
elements of the biset Bpf q of f , see 1.
4.3. Symbolic orbits. We shall be interested in marking periodic orbits of regular
points. These are conveniently encoded in the following simplification of portraits
of bisets (in which the subbisets are singletons and therefore represented simply as
elements):
Definition 4.14 (Symbolic orbits). Let I be a finite index set with self-map
f : I ý, and let B be a G-biset. An I-symbolic orbit is a sequence pbi qiPI of
elements of B, and two I-symbolic orbits pbi qiPI and pci qiPI are conjugate if there
exists a sequence pgi qiPI in G with gi ci “ bi gf piq for all i P I.
△

34

LAURENT BARTHOLDI AND DZMITRY DUDKO

Lemma 4.15. Every homotopy pseudo-orbit can be conjugated to a symbolic orbit,
unique up to conjugacy, in Bpf, A, ordq.
Proof. Given pβi qiPI a homotopy pseudo-orbit, choose paths ℓi from βi p0q to ˚
βi p1q
and define γi “ ℓ´1
. Then γi P Bpf, A, ordq and pβi qiPI „ pγi qiPI .
i #βi #ℓf piq Òf
Furthermore another choice of paths pℓ1i qiPI differs from pℓi qiPI by ℓ1i “ ℓi gi for some
β p1q
gi , so the symbolic orbits pγi qiPI and ppℓ1i q´1 #βi #ℓ1f piq Òf i qiPI are conjugate. 
Let f : pS 2 , Pf , ordf q ý be an expanding map, and let G BG be its biset. Recall
that the Julia set J pf q of f is the accumulation set of preimages of a generic point
˚,
č ď
J pf q :“
f ´m p˚q.
ně0 měn

Every bounded sequence b0 b1 ¨ ¨ ¨ P B b8 defines an element of J pf q as follows:
c
p1q
set c0 “ b0 and ci “ ci´1 #bi Òfi´1
for all i ě 1; then limnÑ8 cn p1q converges
i
to a point ρpb0 b1 ¨ ¨ ¨ q P J pf q. The following proposition directly follows from the
definition:
Proposition 4.16 (Expanding case). Suppose f : pS 2 , Pf , ordf q ý is an expanding
map with orbisphere biset G BG , and let pbi qiPI be a finite symbolic orbit. Let Σ be
a generating set of G BG containing all bi and let ρ : Σ`8 Ñ J pf q be the symbolic
encoding defined above. Then pbi qiPI shadows pρpbi bf piq qbf 2 piq . . . qiPI .

This proposition is useful to solve shadowing problems (namely, determining
when two symbolic orbits shadow the same poly-orbit) using language of automata.
It also follows from the proposition that every orbit homotopy pseudo-orbit shadows a unique orbit in J pf q.
Proposition 4.17 (Shadowing and universal covers). Let f : pS 2 , A, ordq ý be an
r Ñ pS 2 , A, ordq be the universal covering
orbisphere map with biset G BG , let π : G U
2
r be the G equivariant map defined
r
map of pS , A, ordq, and let Φ : G BG b G U ãÑ G U
by Proposition 3.5.
r such that π and Φ extend to continuous
r ` of G U
Then there is a completion G U
`
2
8
r ` with the following property:
r ` Ñ GU
r Ñ S zA and Φ : G BG b G U
maps π : G U
a symbolic orbit pbi qiPI shadows an orbit pxi qiPI in S 2 zA8 if and only if
r ` with πpr
Φpbi b x
rf piq q “ x
ri for some x
ri P U
xi q “ xi .

r` “ U
r.
Furthermore, if ordpaq ă 8 for every a P AzA8 then we may take U

r ` , add to U
r all limit points of parabolic elements corresponding
Proof. To define U
to small loops around punctures a P AzA8 with ordpaq “ 8. The extension of π
r ` zU
r.
and Φ is given by continuity; π is a branched covering with branch locus U
If pbi qiPI shadows pxi qiPI , then there are curves
x
ri : r0, 1s Ñ pS 2 , A, ordq with x
ri p0q “ ˚ and x
ri p1q “ xi
b p1q

such that x
r´1
xf piq Òfi
is a homotopically trivial loop. This exactly means
i #bi #r
that Φpbi b x
rf piq q “ x
ri . Conversely, if Φpbi b x
rf piq q “ x
ri for all i P I then pbi qiPI
shadows pπpr
xi qqiPI .


ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

35

Proposition 4.18 (Shadowing in the p2, 2, 2, 2q case). Using notations of Proposition 4.6, suppose f is a p2, 2, 2, 2q geometric non-invertible map and
Φ:

G BG

b xZ2 ,´z`ry R2 Ñ xZ2 ,´z`ry R2 : pb0 , zq ÞÑ M ´1 z

is as in Proposition 4.6(D). Then for every symbolic orbit pbi qiPI of G BG there is
a unique collection pri qiPI of points in R2 such that ri “ Φpbi b rf piq q. The points
ri are solutions of linear equations (53).
By Proposition 4.17, the image of pri qiPI under the projection π : R2 Ñ R2 {G «
pS , Pf , ordf q is the unique poly-orbit shadowed by pbi qiPI .
The proof will use the following easy fact.
2

Lemma 4.19. If M is non-invertible geometric, then | detpM n ` ǫIq| ě 1 for every
n ě 1 and every ǫ P t˘1u.
Proof. If λ1 , λ2 are M ’s eigenvalues, then Lemma 4.3 gives
| detpM n ` ǫ1q| “ |pλn1 ` ǫqpλn2 ` ǫq| ě 1.


2

Proof of Proposition 4.18. Write bi “ b0 gi . Recall that every gi acts on R as
z ÞÑ ǫi z ` ti with ǫi P t˘1u. We get the following system of linear equations:
M ´1 pǫi rf piq ` ti q “ ri ,

i P I.

By splitting f : I ý into grand orbits and eliminating variables we arrive at equations of the form
(53)

p1 ` θM n qri “ t with θ P t˘1u

and n the period of an orbit and t P R2 some parameters depending on ǫi and ti .
By Lemma 4.19, the system (53) has a unique solution.

Consider an I-symbolic orbit pbi qiPI shadowing a poly-orbit pxi qiPI . This means
(see Definition 4.11) that there are curves pℓi qiPI connecting ˚ to xi such that
bi p1q
ℓ´1
is a trivial loop rel pA, ordq. The local group Gxi ď π1 pX, ˚q “ G
i #bi #ℓf piq Òf
consists of loops of the form
(54)

ℓi r0, 1 ´ εs#α#ℓ´1
i r0, 1 ´ εs,

α is close to xi .

If xi R A, then Gxi is a trivial group; otherwise Gxi is an abelian group of
order ordpxi q. If pA, ordq “ pPf , ordf q, then Gxi is a finite abelian group. Clearly,
pgi bi hi qi also shadows xi for all gi P Gxi and all hi P Gxi`1 . Conversely:
Lemma 4.20. Let f : pS 2 , A, ordq ý be a geometric non-invertible map. Suppose
that the symbolic orbits pbi qiPI and pci qiPI shadow pxi qi . Let Gxi be the local group
associated with pbi qi . Then there are hi P Gxf piq such that pci qiPI is conjugate to
pbi hi qiPI .
If I consists only of periodic indices, then pci qiPI is conjugate to pgi bi qiPI for
some gi P Gxi .
b p1q

i
Proof. By conjugating pci qiPI we can assume that ℓ´1
is a trivial loop; i.e.
i ci ℓi Òf
local groups associated with pci qiPI coincide with whose associated with pbi qiPI . It
is now routine to check that pbi qiPI and pci qiPI differ by the action of local groups.
Indeed, bi is of the form

β p1q

ℓi r0, 1 ´ εs#βi #pℓf piq r0, 1 ´ εsq´1 Òf i

,

βi is close to xi ,

36

LAURENT BARTHOLDI AND DZMITRY DUDKO

and, similarly, ci is of the form
γ p1q

ℓi r0, 1 ´ εs#γi #pℓf piq r0, 1 ´ εsq´1 Òf i

,

γi is close to xi .

Let αi be the image of βi´1 #γi and define hi to be ℓf piq r0, 1 ´ εs#αi#ℓ´1
f piq r0, 1 ´ εs;
compare with (54). Then ci “ bi hi for all i P I.
If I has only periodic indices, then βi´1 and γi end at the same point and we set
αi :“ γi #βi´1 and gi :“ ℓi r0, 1 ´ εs#αi #ℓ´1
i r0, 1 ´ εs. We obtain ci “ gi bi for all
i P I.

Corollary 4.21. Let f : pS 2 , Pf , ordf q ý be a geometric non-invertible map. For
every poly-orbit there are only finitely many symbolic orbits that shadow it. Therefore, there are only finitely many conjugacy classes of portraits in Bpf q.


Theorem 4.22. Let f : pS 2 , A, ordq ý be a non-invertible geometric map. Then
the shadowing operation defines a map from conjugacy classes of symbolic finite
orbits onto poly-orbits in S 2 zA8 . If pA, ordq “ pPf , ordf q, then the shadowing map
is finite-to-one.
Proof. Follows from Propositions 4.13, 4.16, 4.18 and Corollary 4.21.



4.4. From symbolic orbits to portraits of bisets. The centralizer of a symbolic
finite orbit pbi qiPI is the set of pgi qiPI P GI such that gi bi “ bi gf piq for all i P I.

Lemma 4.23. If G BG is the biset of a geometric non-invertible map f and pbi qiPI is
a symbolic
finite orbit shadowing a poly-orbit pxi qiPI , then its centralizer is contained
ś
in iPI Gxi , where Gxi are the local groups associated with pbi qiPI .

Proof. By Definition 4.11 there are curves pℓi qiPI connecting ˚ to xi such that
bi p1q
ℓ´1
is a trivial loop rel pA, ordq. Suppose that pgi qiPI P GI centralizes
i #bi #ℓi Òf
gf piq Òxf i is isotopic to gri rel pA, ordq. By the
pbi qiPI . Set gri :“ ℓ´1
i #gi #ℓi . Then r
expanding property of f , orś
Corollary 4.2 if f is exceptional, gri is trivial rel pA, ordq.

This implies that pgi qiPI P iPI Gxi .

r “
Consider a portrait of bisets pGa , Ba qaPAr in G BG . We may decompose A
n
A \ F \ I with f˚ pF q Ď A for n " 0 and f˚ pIq Ď I. Then for every i P I the group
Gi is trivial and Bi “ tbi u is a singleton. We obtain the symbolic orbit pbi qiPI
which is the essential part of pGa , Ba qAr :
Lemma 4.24. The relative centralizer ZD ppGa , Ba qaPAr q (see §2.8) is isomorphic
to the centralizer of pbi qiPI via the forgetful map pgd qdPD Ñ pgi qiPI .
Let pGa , Ba qaPAr and pG1a , Ba1 qaPAr be two portraits of bisets with associated symbolic orbits pbi qiPI and pb1i qiPI . Assume that Ga “ G1a and Ba “ Ba1 for all a P A.
Then pGa , Ba qaPAr and pG1a , Ba1 qaPAr are conjugate if and only if
(1) pbi qiPI and pb1i qiPI are conjugate; and
(2) G bGd Bd “ G bGd Bd1 for every d P F .

This reduces the conjugacy problem of portrait of bisets to the conjugacy problem of symbolic orbits; indeed Condition (2) is easily checkable. Note that every
r imposes a finite condition on conjugacy and centralizers: for
preperiodic point in A
points attracted to A, Condition (2) imposes a congruence condition modulo the
action on t¨u bG B on the conjugator; for preperiodic points in I, conjugacy again
amounts (by Condition (1) and Definition 4.14) to a congruence condition modulo
the action on t¨u bG B.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

37

Proof. If d P D with f˚ pdq R D, then from gd Bd “ Bd gf˚ pdq “ Bd follows that
gd “ 1. Induction on the escaping time to A gives gd “ 1 for all d P F .
Similarly, if d P D with f˚ pdq R D, then there is a gd P G with gd Bd “ Bd1 “
1
Bd gf˚ pdq if and only if G bGd Bd “ G bGd Bd1 . By induction on the escaping
time, pGa , Ba qaPA\F and pG1a , Ba1 qaPA\F are conjugate if and only if Condition (2)
holds.

We are now ready to show that a geometric map, equipped with a portrait of
bisets, yields a sphere map — possibly with some points infinitesimally close to each
other. A blowup of a two-dimensional sphere is a topological sphere Sr2 equipped
with a monotone map Sr2 Ñ S 2 , namely a continuous map under which preimages
of connected sets are connected. We remark that arbitrary countable subsets of S 2
may be blown up into disks.
r be
Proposition 4.25. Let f : pS 2 , Aq ý be a non-invertible geometric map, let A
r
a set containing a copy of A, let f˚ : A ý be a symbolic map which coincides with
r and let pBa q r be a portrait of bisets in Bpf q.
f on the subset A Ď A,
aPA
r Ñ S 2 extending the identity on A, and a
Then there exists a unique map e : A
2
2
r
blowup b : S Ñ S , with the following properties. The locus of non-injectivity of e
is disjoint from A8 , and b blows up precisely at the grand orbits of
r : x “ epaq “ epa1 qu
tx | Da ‰ a1 P A

r Ă Sr2
replacing points by disks on which the metric is identically 0 and all points in A
2
2 r
r
r
r
are disjoint. The maps f˚ : A ý and f : pS , Aq ý extend to a map f : pS , Aq ý,
which is semiconjugate to f via b, and whose minimal portrait of bisets projects to
pBa qaPAr .

r “ A \ I \ J with
Proof. Let us set G :“ π1 pS 2 zA, ˚q. We may decompose A
f˚ pIq “ I and f˚n pJq Ď A \ I for n " 0. On A, we naturally define e as the identity.
If i P I, then i is f˚ -periodic and the bisets Bi , Bf˚ piq , . . . determine a homotopy
pseudo-orbit which shadows a unique periodic poly-orbit in S 2 , by Theorem 4.22.
Thus e is uniquely defined.
We now blow up the grand orbits of all points in S 2 which are the image of more
than one point in A \ I under e, replacing them by a disk on which the metric
is identically 0. We inject A \ I arbitrarily in the blown-up sphere Sr2 , and now
identify A \ I with its image in the blowup.
Let us next extend f to a self-map fr of Sr2 so that pBa qaPA\I is the induced
portrait of bisets. We first do it by arbitrarily mapping the disks to each other by
homeomorphisms restricting to f˚ on A \ I. Let pBa1 qaPA\I be the projection of
the minimal portrait of bisets of fr via π1 pSr2 , A \ Iq Ñ G. Consider i P I. By
Lemma 4.20 (the periodic case), we can assume that bi “ hi b1i with hi P Gepiq .
(Note that if epiq R A, then hi “ 1 and bi “ b1i .) Let m1 P BraidpSr2 zA, Iq be a
preimage of phi qiPI under EI (see (19)) and set m :“ pushpm1 q; it is defined up to
pre-composing with a knitting element. Since hi P Gepiq , we can assume that m
is identity away from the blown up disks. Then pBa qaPA\I is a portrait of bisets
induced by mfr.
Consider next j P J and assume that f˚ pjq has already been defined. The biset
Bj , and more precisely already its image in t¨u bG Bpf q, see Definition 2.15(B),

38

LAURENT BARTHOLDI AND DZMITRY DUDKO

determines the correct fr-preimage of f˚ pjq P Sr2 that corresponds to j, and thus
determines epjq uniquely.


4.5. Promotions of geometric maps. Let X :“ pS 2 , A, ordq be an orbisphere,
and consider a subset D Ă S 2 zA. Recall from (11) the quotient ModpX|Dq of
mapping classes of pS 2 , A \ Dq by knitting elements; there, we called two maps
f, g : pX, Dq ý knitting-equivalent, written f «A|D g, if they differ by a knitting
element in knBraidpX, Dq. We shall show that knitting-equivalent rigid maps are
conjugate rel A \ D:
Theorem 4.26 (Promotion). Suppose f, g : pX, Dq ý are orbisphere maps, and
assume that either
‚ g is geometric non-invertible, or
‚ g m pDq Ď A for some m ě 0.
Then every conjugacy h P ModpX|Dq between f and g rel A|D lifts to a unique
conjugacy r
h P ModpS 2 , A \ Dq between f and g rel A \ D such that r
h «A|D h.
Proof. We begin by a

Lemma 4.27. Under the assumption on g, consider that b P BraidpS 2 zA, Dq. If
for every m ě 1 the element b is liftable through g m , then pg ˚ qm b is trivial for all
m ą log |g|, where | | is the word metric.
Proof. Consider first the case that g is expanding. Then the lengths of curves
pg ˚ qm pbqp´, aq tend to 0 as m Ñ 8, so pg ˚ qm pbq “ 1 for sufficiently large m. If
g is exceptional, then Corollary 4.2 replaces the expanding argument. If finally
g m pEq Ď A, then pg ˚ qm pbq “ 1 for the same m.


We resume the proof. Let r
h0 P pModpS 2 , A \ Dq be any preimage of h under
2
the forgetful map ModpS , A \ Dq Ñ ModpX|Dq. Then r
h0 f r
h´1
«A\D h´1
0
1 g
´1
r
r
r
r
for some h1 P knBraidpX, Dq. Setting h1 :“ h1 h0 we get h1 f h1 «A\D h´1
2 g,
´1
is
the
lift
of
h
through
g.
Continuing
this
process
we
eventually
get
where h´1
2
1
´1
´1
r
r
hm f hm «A\D g because a the corresponding lift of h1 is trivial by Lemma 4.27.
We have shown the existence of r
h.
If r
h1 is another promotion of h, then r
h1 and r
h differ by a knitting element
h1 “
commuting with f . By Lemma 4.27, that knitting element is trivial, hence r
r
h.


Corollary 4.28. Let ModpS 2 ,A\Dq M pf qModpS 2 ,A\Dq Ñ ModpX|Dq M pf |DqModpX|Dq
be the inert map forgetting the action of knitting elements (see Proposition 2.12).
Suppose that either f is geometric non-invertible or f m pDq Ď A for some m ě 0.
Then f, g are conjugate in M pf q if and only if their images are conjugate in
M pf |Dq. Moreover, the centralizers Zpf q and Zpf |Dq (see §2.8) are naturally
isomorphic via the projection ModpS 2 , A \ Dq Ñ ModpX|Dq.

4.6. Automorphisms of bisets. Recall that the automorphism group of a biset
H BG is the set AutpBq of maps τ : B ý satisfying τ phbgq “ hτ pbqg for all h P
H, g P G.
Proposition 4.29. If H BG is a left-free, right-transitive biset and H is centreless,
then AutpBq acts freely on B, so B{ AutpBq is also a left-free, right-transitive HG-biset.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

39

Proof. AutpBq acts by permutations on t¨u bH B, and commutes with the right
G-action, which is transitive, so AutpBq{ kerpactionq acts freely. Consider now
τ P AutpBq that acts trivially on t¨u bH B, and consider b P B. We have τ pbq “ tb
for some t P H, and for all h P H there exists g P G with hb “ bg, because B is
right-transitive. We have
htb “ hτ pbq “ τ phbq “ τ pbgq “ τ pbqg “ tbg “ thb,

so t P ZpHq “ 1 and therefore τ “ 1. It follows that AutpBq acts freely on B.
Corollary 4.30 (No biset automorphisms). For an orbisphere biset
is non-cyclic, we have AutpBq “ 1.
In the dynamical situation

G BG

H BG



with H

is a cyclic biset if and only if G is a cyclic group.

Proof. Since B is an orbisphere biset, it is left-free and right-transitive; and since it
is not cyclic, H is a non-abelian orbisphere group and in particular is centreless, so
Proposition 4.29 applies. Now B cannot have a proper quotient, because conjugacy
classes of elements of H appear exactly once in wreath recursions of peripheral
elements, by [5, Definition 2.6(SB3 ) of orbisphere bisets].

Corollary 4.31. Suppose f, g : pS 2 , C, ordq Ñ pS 2 , A, ordq are isotopic orbisphere
maps. Let Bpf q and Bpgq be the bisets of f and g with respect to the same base
points : P S 2 zC and ˚ P S 2 zA. Then there is a unique isomorphism between Bpf q
and Bpgq.
Proof. Consider an isotopy pft qtPr0,1s rel A from f to g. It induces a continuous motion Bpft , A, ord, ˚q of the bisets of ft . Therefore, all Bpft , A, ord, ˚q are isomorphic.

This shows existence, the uniqueness follows from Corollary 4.30.
Theorem 4.32 (Rigidity). Suppose f : pS 2 , A, ordq ý is a geometric map. If
h : pS 2 , Aq ý commutes with f , is the identity in some neighbourhood of A8 , and
is isotopic to 1 rel A, then h “ 1.

Proof. Fix a basepoint ˚ P S 2 zA and the fundamental group G “ π1 pS 2 zA, ˚q.
Since h is isotopic to the identity rel A, there is a path ℓ : r0, 1s Ñ S 2 zA from ˚ to
hp˚q such that γ « ℓ#ph ˝ γq#ℓ´1 for all γ P G. Define then
h˚ :

G Bpf qG

ý,

hpβp1qq

h˚ pβq :“ ℓ#ph ˝ βq#pℓ´1 qÒf

.

Since h commutes with f , this defines an automorphism of Bpf q. By Corollary 4.30,
it is the identity on Bpf q. It also fixes, therefore, all conjugacy classes in Bpf qbn
for all n P N. By Theorem 4.22, these are in bijection with periodic orbits of f . It
follows that h is the identity on all periodic points of f , and therefore on f ’s Julia
set. Since furthermore h is the identity near A8 , and every point of S 2 zA either
belongs to J pf q or gets attracted to A8 , we get h “ 1 everywhere.

4.7. Weakly geometric maps. We are now ready to show that maps whose restriction to their minimal orbisphere is geometric are of a particularly simple form.
Definition 4.33 (Tunings). Let f : pS 2 , Aq ý be a Thurston map. A tuning
multicurve for a f is an f -invariant1 multicurve C such that f ´1 pC q does not
contain nested components rel f ´1 pAq; namely, the adjacency graph of components
of S 2 zf ´1 pC q is a star.
1In the sense that f ´1 pC q equals C rel A.

40

LAURENT BARTHOLDI AND DZMITRY DUDKO

A tuning is an amalgam of orbisphere bisets in which the graph of bisets is a star.
Equivalently, it is an amalgam of Thurston maps along a tuning multicurve.
△
Every tuning has a central map, corresponding to the centre of the star, as well
as satellite maps, corresponding to its leaves. Furthermore, unless the star has two
vertices, its central map is uniquely determined.
Definition 4.34 (Weakly geometric maps). A tuning by homeomorphisms is a
tuning in which all satellite maps are homeomorphisms.
A map is weakly geometric if it is a (possibly trivial) tuning by homeomorphisms
in such a manner that the central map is isotopic to a geometric map.
△
Theorem 4.35. Let f : pS 2 , Aq ý be a non-invertible Thurston map. Then f is
weakly geometric if and only if its restriction f : pS 2 , Pf , ordf q ý is isotopic to a
geometric map.
Proof. If f is weakly geometric, then its central map f is isotopic to a geometric
map, and f is isotopic to f rel Pf .
Conversely, assume that the restriction f : pS 2 , Pf , ordf q ý of f is geometric,
and consider its portrait of bisets pBa qaPA induced from the minimal portrait of
bisets of f . By Proposition 4.25, there exists a blown-up sphere Sr2 and extension
fr of f , such that the portraits of bisets of f and fr are conjugate portraits of bisets
in Bpf q. The tuning multicurve we seek is the boundary of the blowup disks.
By Theorem 2.19, the bisets Bpf q and Bpfrq differ by a knitting element, so
r
we may write fr “ mf for some m P knBraidpS 2 zA, AzAq.
By Proposition 2.8,
the class m is liftable arbitrarily often through f , and by Lemma 4.27 its lift
becomes eventually trivial; in other words, we have a relation n´1 frn “ mf for
some m with trivial image in BraidpS 2 zAq, and therefore n is a product of mapping
classes in the infinitesimal disks. Its restriction to these disks yields the required
homeomorphisms of the tuning.

We now turn to the algebraic aspects of geometric maps.
Definition 4.36. An orbisphere biset is geometric if it is the biset of a geometric
map. A biset B is weakly geometric if its projection B to its minimal orbisphere
group is geometric.
△
Corollary 4.37. A Thurston map f is [weakly] geometric if and only if its biset
is [weakly] geometric.

The following result was already obtained by Selinger and Yampolsky in the case
of torus maps [23]:
Corollary 4.38. A non-invertible Thurston map is isotopic to a geometric map if
and only if it is Levy-free.
Proof. Let f be a non-invertible Thurston map. If f admits a Levy cycle, then
either f shrinks no metric under which a curve in the Levy cycle is geodesic, or f
is doubly covered by a torus endomorphism with eigenvalue ˘1; in all cases, f is
not geometric.
Conversely, if f is Levy-free, then its restriction f to pS 2 , Pf , ordf q is still Levyfree. By [4, Theorem A] (for tExpu maps) or Lemma 4.4 (for tGTor/2u maps) the
map f is geometric. By Proposition 4.25, there is a map e : A Ñ S 2 whose image is

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

41

preserved by f . If e were not injective, there would be a Levy cycle for f consisting
of curves surrounding elements of A with same image under e. Now if e is injective

then the tuning constructed in Theorem 4.35 is trivial, so f is geometric.
4.8. Conjugacy and centralizer problem. We are now ready to show how conjugacy and centralizer problems may be solved in geometric bisets. We review the
algebraic interpretation of expanding maps: by [4, Theorem A], a map is expanding if and only if its minimal biset is contracting. We recall briefly that a left-free
G-G-biset B is contracting if, for every basis X Ď B, there exists a finite subset
N Ď G with the following property: for every g P G there is n P N such that
X n g P N X n Ď B bn .
The minimal such subset N is called the nucleus associated with pB, Xq. Note
that different bases yield different nuclei, but that finiteness of N is one basis implies
its finiteness for all other bases, or even for all finite sets X generating B qua left
G-set.
The biset B may be given by its structure map X ˆ G Ñ G ˆ X, written
px, gq ÞÑ pg@x, xg q; it is defined by the equality xg “ pg@xqxg in B. Then B is
contracting with nucleus N if for every g P G every sufficiently long iteration of the
maps g ÞÑ g@x eventually gives elements in N .
We may assume, without loss of generality, that XN Ď N X holds. For finitely
generated groups, there is perhaps more intuitive formulation of contraction: there
exists a proper metric on G and constants λ ă 1, C such that }g@x} ď λ}g} ` C
for all g P G, x P X.
Lemma 4.39. Let G BG be a contracting biset; choose a basis X of B, and let
N Ď G be the associated nucleus.
Then every symbolic finite orbit pbi qiPI is conjugate to one in which bi P N X for
all i P I.
Proof. Write every bi in the form gi xi with gi P G and xi P X. Conjugate the
symbolic orbit by pgi qiPI ; then each bi becomes gi´1 bi gf piq “ xi gf piq “ gi1 x1i P B for
some gi1 P G, x1i P X. Note that each gi1 is a state of some gi . Conjugate again by
pgi1 qiPE , etc.; after a finite number of steps, each bi will belong to N X.

Lemma 4.40. Let f : pS 2 , A, ordq ý be a geometric map. Then for every n P N
the number of n-periodic points of f is finite.
Proof. If f is doubly covered by a torus endomorphism z ÞÑ M z `q, then its periodn points are all solutions to z P p1 ´ M q´1 q ` p1 ´ M n q´1 Z2 ; the image of this set
under R2 Ñ R2 {xZ2 , ´zy is finite, of size at most detp1 ´ M n q.
If f is expanding, then consider its biset B, which is orbisphere contracting,
see [4, Theorem A]. By Lemma 4.39, there are finitely many conjugacy classes of
period-n finite symbolic orbits in B, and by Theorem 4.22, every symbolic orbit
shadows a unique finite poly-orbit.

r Ñ G be a forgetful morphism of groups and let
Theorem 4.41. Let G
r Ñ G BG and

r BG
r
G

r1
r BG
r
G

Ñ GB1G

r is geometbe two forgetful biset morphisms as in (8). Suppose furthermore that B
ric of degree ą 1. Denote by pGa , Ba qaPAr and pG1a , Ba1 qaPAr the portraits of bisets
r and B
r 1 in B and B 1 respectively.
induced by B

42

LAURENT BARTHOLDI AND DZMITRY DUDKO

r B
r 1 are conjugate under ModpGq
r if and only if there exists φ P ModpGq
Then B,
φ
1
φ
such that B – B and the portraits pGa , Baφ qaPAr and pG1a , Ba1 qaPA\E are conjugate.
Furthermore, the centralizer of the portrait pGa , Ba qaPAr is trivial, and the cenr of B
r is isomorphic, via the forgetful map ModpGq
r Ñ ModpGq, to
tralizer ZpBq
ˇ φ φ
(
φ P ZpBq ˇ pGa , Ba q r „ pGa , Ba q r
aPA

aPA

and is a finite-index subgroup of ZpBq.

r B
r 1 are conjugate, then certainly their images and subbisets are conProof. If B,
jugate. Conversely, let φ P ModpGq be such that B φ – B 1 and the portraits
r B
r1
pGφa , Baφ qaPAr and pG1a , Ba1 qaPA\E are conjugate, and let fr, fr1 be maps realizing B,
respectively, with f geometric. By Theorem 2.19, we have fr1 “ mfr for a knitting
mapping class m. Since m is liftable by Proposition 2.8, we have mfr “ f m1 for
some knitting mapping class m1 and therefore m1 fr1 pm1 q´1 “ m1 fr. Repeating with
m1 , we obtain mpkq ¨ ¨ ¨ m1 fr1 pmpkq ¨ ¨ ¨ m1 q´1 “ mpkq fr for all k P N, and mpkq “ 1
when k is large enough by Lemma 4.27. Therefore, fr, fr1 are conjugate, and so are
r B
r1.
B,
r follows. The centralizer of the portrait pGa , Ba q r
The description of ZpBq
aPA
r is geometric) all points shadowed by bisets Ba are
is trivial, because (since B
unmarked in G. The ZpBq-orbit of pGa , Ba qaPAr contains finitely many conjugacy
classes by Lemma 4.40, since all of them are induced from a geometric biset, and
therefore are encoded in periodic or pre-periodic points for a map realizing B. 
5. Algorithmic questions
We give some algorithms that decide whether two portraits of bisets are conjugate, thereby reducing the conjugacy and centralizer problem from orbisphere maps
to their minimal orbisphere (with only the post-critical set marked).
All the algorithms we describe make use of the symbolic description of maps via
bisets, and are inherently quite fast and practical. Their precise performance, and
implementation details, be studied in the last paper [6] of this series.
There are two implementations of Algorithms 1.1 and 1.2, one for tGTor/2u
maps and one for tExpu maps. We describe them in separate subsections.
We already gave the following algorithms:
Algorithm 5.1 ([4, Algorithms 5.1 and 5.2]). Given an orbisphere biset G BG ,
Decide whether B is the biset of a map double covered by a torus endomorphism,
and if so Compute parameters M, q for a torus endomorphism z ÞÑ M z`q covering
B.
Algorithm 5.2 ([4, Algorithms 5.4 and 5.5]). Given an orbisphere biset G BG ,
Decide whether B is the biset of a tGTor/2u or an tExpu map, and in particular
whether B is geometric.
5.1. tGTor/2u maps. In the case of tGTor/2u maps, we can decide, without access
to an oracle, whether two such maps are conjugate, and we can compute their
centralizers, as follows. We shall need the following fact:
Theorem 5.3 (Corollary of [10]). There is an algorithm deciding whether two
matrices M, N P Mat`
2 pZq are conjugate by an element X P SL2 pZq, and produces
such an X if it exists.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

43

There is an algorithm computing, as a finitely generated subgroup of SL2 pZq, the
centralizer of M P Mat`

2 pZq.
Algorithm 5.4. Given G BG , G CG two minimal tGTor/2u bisets
Decide whether B and C are conjugate by an element of ModpGq, and if so construct a conjugator; and compute the centralizer ZpBq, as follows:
(1) If B˚ ‰ C˚ as maps on peripheral conjugacy classes, then return fail.
(2) Using Proposition 4.5(A), identify G with Z2 ¸ t˘1u, and with Algorithm 5.1 present B and C as BM v and BN w respectively, see (50).
(3) Using Theorem 5.3 check whether M and N are conjugate. If not, return
no; otherwise find a conjugator X and compute the centralizer ZpM q of M .
(4) Check whether there is a Y P ZpM q such that BpY Xq0 conjugates BM v to
BN w , as follows. The orbit of BM v under conjugation of
tBY 0 | Y P ZpM q, BpY Xq0 P ModpGqu

is finite and hence computable; so is its image under X 0 . Check whether
BN w belongs to it; if not, return no, and if yes, return yes and the conjugator BpY Xq0 .
(5) The centralizer of B is
tBY 0 | Y P ZpM q, BY 0 P ModpGq, and BY 0 centralizes BM v u,

naturally embeds as a subgroup of finite index in ZpM q.
Here is an algorithmic version of Proposition 4.18:

Algorithm 5.5. Given a minimal tGTor/2u orbisphere biset
xZ2 , ´z ` ry, a base point b0 P B specifying the map
(55)

Φ:

G BG

G BG

with G “

b xZ2 ,´z`ry R2 Ñ xZ2 ,´z`ry R2 : pb0 , zq ÞÑ M ´1 z

rÑA
r of the dynamics of B on its pe(see Proposition 4.18), an extension f˚ : A
ripheral classes, and a portrait of bisets pGa , Ba qaPAr ,
Compute pra qaPAr shadowed by pGa , Ba qaPAr : ra “ ΦpBa b rf˚ paq q, the local groups
Gr̄a where r̄a is the image of ra in R2 {G (see (54)), and the relative centralizer
ZD ppGa , Ba qaPAr q, which is a finite abelian group, as follows:
(1) Choose ba P Ba . Proceed through all periodic cycles E of I. Choose e P E
and solve the linear equation
´
¯
re “ Φ|E| be b bf˚ peq b ¨ ¨ ¨ b bf |E|´1 peq b re ;
˚

n

(2)
(3)
(4)
(5)

the equation takes form p1 ` θM qri “ t with θ P t˘1u (see (53)) and has
a unique solution by Lemma 4.19.
r
Inductively compute ra “ Φpba b rf˚ paq q for all a P A.
For a P A we have Gr̄a “ Ga .
r check whether ra ´ z P G. If ra ´ z R G, then Gr̄a is the trivial
For a P AzA
subgroup; otherwise Gr̄a “ xra ´ zy.
By a finite check compute ZD ppGa , Ba qaPAr q: by Lemma 4.24 it is the set
of self-conjugacies of the corresponding
homotopy pseudo-orbit, and by
ś
Lemma 4.20 it is a subgroup of aPAr Gr̄a , and is therefore an easily computable finite group.

44

LAURENT BARTHOLDI AND DZMITRY DUDKO

Algorithm 5.6. Given a minimal tGTor/2u orbisphere biset G BG , an extension
rÑ A
r of the dynamics of B on its peripheral classes, and two portraits of
f˚ : A
bisets pGa , Ba qaPAr and pG1a , Ba1 qaPAr ,
Decide whether pGa , Ba qaPAr and pG1a , Ba1 qaPAr are conjugate as follows:

(1) Normalize the portraits in such a manner that Ga “ G1a and Ba “ Ba1 for
all a P A; by Lemma 2.17 this follows from the conjugacy for subgroups:
find pℓa qaPA P GA with Gℓaa “ G1a and conjugate pGa , Ba qaPAr by pℓa qaPAr
with ℓa “ 1 if a R A.
(2) Identify G with xZ2 , ´z ` ry and choose b0 P B characterizing the map Φ,
see (55).
(3) Using Algorithm 5.5 compute the points ra and ra1 shadowed by pGa , Ba qaPAr
and pG1a , Ba1 qaPAr respectively.
r then return no.
(4) Check if r̄a “ r̄a1 in R2 {G. If r̄a ‰ r̄a1 for some a P A,
1
Otherwise find ℓa P G with ℓa ra “ ra with ℓa “ 1 for a P A and conjugate
pG1a , Ba1 q by pra qaPAr . This reduces to original problem to the case ra1 “ ra .
(5) Using Algorithm 5.5 compute the (finite
ś abelian) local groups Gr̄a and by
a finite check decide if an element in aPAr Gr̄a conjugates pGa , Ba qaPAr to
pG1a , Ba1 qaPAr .

r
and
If pr̄a qaPAr is an actual orbit, then Gr̄a are trivial groups for all a R AzA
Step (5) can be omitted. This is the case when the algorithm is called from Algorithm 5.12.

Algorithm 5.7. Given a minimal tGTor/2u orbisphere biset G BG and an extenrÑA
r of the dynamics of B on its peripheral classes,
sion f˚ : A
Produce a list of all conjugacy classes of portraits of bisets pGa , Ba qaPAr in B
with dynamics f˚ as follows:
(1) Write G “ xZ2 , ´z ` ry and B “ BM v , using Algorithm 5.1. Choose b0 P B
characterizing the map Φ, see (55).
(2) Using Algorithm 5.5 compute the orbit pr̄a qaPA of M : R2 {G ý. Produce a
list of all possible poly-orbits pr̄a qaPAr extending pr̄a qaPA .
(3) For every poly-orbit pr̄a qaPAr find a portrait pGa , Ba qaPAr that shadow pr̄a qaPAr .
(4) Using Algorithm 5.5 compute the (finite) local groups Gr̄a . By Lemma 4.20
the finite set
tpGa , Ba ha qaPAr | ha P Gr̄a , ha “ 1 if a P Au

contains a representative of every conjugacy class of portraits that shadows
pr̄a qaPAr . Using Algorithm 5.5 produce a list of all conjugacy classes of
portraits of bisets that shadow pr̄a qaPA .
5.2. tExpu maps. We now turn to expanding maps, and start by a short example
showing how the conjugacy problem may be solved algorithmically. We note, by
following the proof of Lemma 4.39, that every portrait of bisets can be algorithmically conjugated to one in which the terms belong to N X, for X a basis and N the
associated nucleus.
Example 5.8. Suppose E “ teu with f˚ peq “ e, and let pGa , Ba qaPA\E and
pGa , Ca qaPA\E be two portraits of bisets. Then Ge “ 1 and Be “ tbu and Ce “ tcu.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

45

The portraits pGa , Ba qaPA\E and pGa , Ca qaPA\E are conjugate if and only if
there exists ℓ P G such that ℓ´1 bℓ “ c.
Write Be “ tgxu in a basis X of B, with associated nucleus N ; recall that N is
symmetric, contains 1 and generates G. Then gx is conjugate to xg “ pg@xqxg “
g 1 x1 . After iterating finitely many times the process gx „ xg “ g 1 x1 , we can assume
g P N , as in Lemma 4.39. Similarly, we may replace Ce by a conjugate biset thyu
with h P N .
Find, by direct search, a t P N with N t´3 X Ě XN t ; such a t ě 4 exists because
B is contracting. Then Be and Ce are conjugate if and only if there exists ℓ P G
with bℓ “ ℓc, namely gxℓ “ ℓhy. Let u P N be such that ℓ P N u zN u´1 ; if u ě t
then gxℓ P N N u´3 X while ℓhy R N u´1 N X, a contradiction. Therefore the search
for a conjugator ℓ is constrained to ℓ P N t´1 .
As in Example 5.8 we have
Lemma 5.9. Let G BG be a contracting biset; choose a basis X of B, and let N Ď G
be the associated nucleus. Suppose that t P N is such that N t´3 X Ě XN t .
If two symbolic orbits pbi qiPI and pci qiPI with bi , ci P N X are conjugate by gi ,
then gi P N t´1 .
Proof. Suppose gi P N u zN u´1 for u ě t and some i. We can also assume that
gj P N u for all j P I. Then on one hand, gi bi P N u N X; but on the other hand
gi bi “ ci gf˚ piq P N u´3 N X. This is a contradiction.

Algorithm 5.10. Given a minimal tExpu orbisphere biset G BG , an extension
rÑ A
r of the dynamics of B on its peripheral classes, and two portraits of
f˚ : A
bisets pGa , Ba qaPAr and pG1a , Ba1 qaPAr ,
Decide whether pGa , Ba qaPAr and pG1a , Ba1 qaPAr are conjugate, and Compute the
centralizer of pGa , Ba qaPAr , which is a finite group, as follows:

r “ A\J \I with f n pJq Ď A and f˚ pIq Ď I. Normalize pGa , Ba q r
(1) Write A
˚
aPA
and pG1a , Ba1 qaPAr such that Ga “ G1a and Ba “ Ba1 ; by Lemma 2.17 this
follows from the conjugacy for subgroups: find pℓa qaPA P GA with Gℓaa “ G1a
and conjugate pGa , Ba qaPAr by pℓa qaPAr with ℓa “ 1 if a R A.
(2) Check whether there is pℓa qaPA\J with ℓa “ 1 for a P A conjugating
pGa , Ba qaPA\J and pG1a , Ba1 qaPA\J . If not, return no. This reduces the
conjugacy problem of portraits to the conjugacy problem of symbolic orbits: writing Bi “ tbi u and Bi1 “ tb1i u for i P I solve a conjugacy problem
of pbi qiPI and pb1i qiPI .
(3) Write the biset B in the form GX for a basis X, and let N be its nucleus.
Find, by direct search, a t P N with N t´3 X Ě XN t ; such a t ě 4 exists
because B is contracting.
(4) Write bi “ gi xi and replace bi with xi gi “ gi1 x1i . After iterating finitely
many times this process, we obtain bi P N X by Lemma 4.39. By a similar
iteration, we can assume b1i P N X.
(5) Answer whether pbi qiPI and pb1i qiPI are conjugate by elements in N t´1 . This
is correct by Lemma 5.9.
(6) By a direct search compute the centralizer of pGa , Ba qaPAr : the centralizer of
pGa , Ba qaPA\J is trivial, while elements centralizing pGa , Ba qaPI are within
N t´1 .

46

LAURENT BARTHOLDI AND DZMITRY DUDKO

Algorithm 5.11. Given a minimal tExpu orbisphere biset G BG and an extension
rÑA
r of the dynamics of B on its peripheral classes,
f˚ : A
Produce a list of all conjugacy classes of portraits of bisets pGa , Ba qaPAr in B
with dynamics f˚ as follows:
r “ A \ J \ I with f n pJq Ď A and f˚ pIq Ď I. Produce a list LA\J
(1) Write A
˚
of all conjugacy classes of portraits of bisets pGa , Ba qaPA\J .
(2) Write the biset B in the form GX for a basis X, and let N be its nucleus. Produce a list of all symbolic orbits pbi qiPI with bi P N X. Using
Algorithm 5.10 produce a list LI of all conjugacy classes of symbolic orbits
pbi qiPI with bi P N X.
(3) Combine LA\J and LI to produce a list LA\J ˆ LI of all conjugacy classes
of portraits of bisets pGa , Ba qaPAr by setting Bi – tbi u and Gi – t1u for
i P I.
5.3. Decidability of conjugacy and centralizer problems. The statements of
Theorem E can be turned into an algorithm:
r r, rC
r r two geometric bisets,
Algorithm 5.12. Given Gr B
G G G
And given an oracle that decides whether two minimal tExpuztGTor/2u bisets are
conjugate, and computes their centralizers,
r and C
r are conjugate by an element of ModpGq,
r and if so conDecide whether B
r
struct a conjugator; and compute the centralizer ZpBq, as follows:

r˚ ‰ C
r˚ as maps on peripheral conjugacy classes, then return no.
(1) If B
r
r r Ñ G1 CG1 be the maximal forgetful morphisms.
(2) Let Gr BGr Ñ G BG and Gr C
G
1
If G ‰ G , then return no.
(3) From now on assume G “ G1 . Using Algorithm 5.4 or the oracle, decide
whether B, C are conjugate; if not, return no. Using Algorithm 5.4 or the
oracle compute the centralizer ZpBq.
(4) From now on, assume B φ “ C for some φ P ModpGq. Compute the action
of ZpBq on the list of all conjugacy classes of portraits of bisets (this list
is finite by Corollary 4.21; Algorithms 5.7 and 5.11 compute the list) and
´1
´1
check whether pGa , Ba qaPAr and pGφa , Caφ qaPAr belong to the same orbit.
Return no if the answer is negative. Otherwise replace φ by an element in
ZpBqφ such that pGφa , Baφ qaPAr and pGa , Ca qaPAr are conjugate portraits.
r of φ, and compute k P ModpGq
r
(5) Choose an arbitrary lift φr P ModpGq
r
φ
r – C.
r It follows that k is a knitting element. Compute
such that k B
r 1 k2 ¨¨¨kpn´1q
pnq
r φk
r proceed until k pnq is identity
inductively k
with k pnq B
– C;
r 1 k 2 ¨ ¨ ¨ k pn´1q .
(this is guaranteed by Lemma 4.27). Return φk
r
(6) To compute the centralizer of B, consider first ZpBq. Compute the action
of ZpBq on the list of all conjugacy classes of portraits of bisets (this list
is finite by Corollary 4.21; Algorithms 5.7 and 5.11 compute the list) and
compute the stabilizer Z0 pBq of the ZpBq-action. Note that Z0 pBq is a
finite-index subgroup of ZpBq.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

47

(7) List a generating set S of Z0 pBq. For every φ P S compute its lift φp P
r with B
r ψp – B:
r compute first an arbitrary lift φr of φ, then inducModpGq
r 1 k2 ¨¨¨kpn´1q
pnq
r φk
r until k pnq is identity, and set
tively define k
with k pnq B
–B
1
2
pn´1q
r k ¨¨¨k
φp :“ φk
. Return tφp | φ P Su.

Proof of Corollary F. We claim that Algorithms 5.5, 5.6, 5.7, 5.10, 5.11 are efficient.
Indeed, the efficiency of Step 4 of Algorithm 5.10 follows from expansion. The
efficiency of Steps 5 and 7 of Algorithm 5.12 follows from Lemma 4.27. All the
remaining steps are obviously efficient.


We note that, in the case of p2, 2, 2, 2q bisets, the oracle itself is efficient. Indeed the oracle reduces to solving conjugacy and centralizer problems in the group
SL2 pZq, and Theorem 5.3 is efficient, because SL2 pZq has a free subgroup of index
12.
Corollary 5.13. There is an algorithm that, given two geometric bisets G BG and
decides whether B and C are conjugate, and computes the centralizer of B.

H CH ,

Proof. We briefly sketch an algorithm that justifies an existence of an oracle: given
two minimal tExpu orbisphere bisets, whether they are conjugate and computes
their centralizers; details will appear in [6].
Let B, C be two minimal tExpu orbisphere G-G-bisets. They admit a decomposition into rational maps along the canonical obstruction, which is computable
by [22]. The graph of bisets along this decomposition is computable, and rational maps may be computed for each of the small bisets in the decomposition, e.g.
by giving their coëfficients as algebraic numbers with floating-point enclosures to
distinguish them from their Galois conjugates. The bisets B, C are conjugate precisely when their respective rational maps are conjugate and the twists along the
canonical obstruction agree; the first condition amounts to finite calculations with
algebraic numbers, while the second is the topic of [5, Theorem A].
The centralizer of a rational map is trivial, and [5, Theorem A] shows that the
centralizer of B is computable.

Proof of Corollary G. If the rational map is p2, 2, 2, 2q, then the oracle is efficient,
as we noted above, so Corollary F applies. In the other case, the rational map is
hyperbolic, so it has trivial centralizer and no oracle is needed in the application of
Algorithm 5.12.

6. Examples
Finally, in this brief section, we consider some examples of portraits of bisets.
The first ones come from marking points on the Basilica map f pzq “ z 2 ´ 1, and
more generally maps with three post-critical points. The second ones from cyclic
bisets (which are particularly simple, but to which our main results apply with
restrictions because these bisets have automorphisms).
6.1. Twisted marked Basilica. Consider the Basilica polynomial
p t0, ´1, 8uq ý .
f pzq :“ z 2 ´ 1 : pC,

It has two fixed points α and β, with α P p´1, 0q and β ą 1. Let us take α to be
the basepoint and let G BG be the biset of f . Denote respectively by γ´1 and γ0

48

LAURENT BARTHOLDI AND DZMITRY DUDKO

γ´1 ´1 ‚

α
‚

γ0

´α
‚

‚ 0

‚

β

x2
γβ

t

Figure 1. The dynamical plane of z 2 ´ 1. Loops γ´1 , γ0 , γβ circle around 0, ´1, β respectively. The curve x2 connects α to its
preimage ´α. The simple closed curve t surrounds t´1, βu.
the loops circling around ´1 and 0 as in Figure 1, and let γ8 “ pγ´1 γ0 q´1 be the
loop around infinity. Then
G “ xγ´1 , γ0 , γ8 | γ8 γ´1 γ0 y.
The basepoint α has two preimages α and ´α. Let x1 be the constant path at
α and let x2 be a path slightly below 0 connecting α to ´α. The presentation of
B in the basis S :“ tx1 , x2 u is
γ´1 “!1, γ0 "p1, 2q,
γ0 “!γ´1 , 1",

´1 ´1
γ8 “!γ´1
γ0 , 1"p1, 2q.

We shall consider first the effect of marking a fixed point, and then of marking
some preimages of the post-critical set.
p t8, 0, ´1, βuq ý.
6.1.1. Marking β. Consider a marked Basilica frpzq “ z 2 ´ 1 : pC,
Let γβ be the loop around β depicted in Figure 1. The fundamental group of
p t8, 0, ´1, βuq, based at α, is
pC,
r “ xγ´1 , γ0 , γβ , γ8 | γ8 γ´1 γβ γ0 y
G

r ։ G sends γβ to 1. The presentation of the biset B
r “ Bpfrq
and the forgetful map G
in the basis S is
γ´1 “!1, γ0 "p1, 2q,
γβ “!1, γβ ",

γ0 “!γ´1 , 1",

´1 ´1
γ8 “!γ´1
γ0 , γβ´1 "p1, 2q.

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

49

r “ t8, ´1, 0, βu; then B
r has a minimal portrait of
Write A “ t8, ´1, 0u and A
bisets
r ´1 “ xγ´1 y,
G

r´1 “ G
r´1 x1 ,
B

rβ “ xγβ y,
G

rβ “ G
r´1 x2 ,
B

r ´1 ,
r0 “ G
r0 x1 \ G
r0 x2 “ x1 G
B

r 0 “ xγ0 y,
G

r8 “ xγ8 y,
G

r8 “ G
r8 pγ´1 x1 q \ G
r8 pγ´1 x2 γβ q.
B

r ։ B the portrait pG
ra , Ba q r projects to
Under the forgetful intertwiner B
aPA
G´1 “ xγ´1 y,

B´1 “ G´1 x1 ,

Gβ “ 1,

Bβ “ tx2 u,

G0 “ xγ0 y,

B0 “ G0 x1 \ G0 x2 “ x1 G´1 ,

G8 “ xγ8 y,

B8 “ G8 pγ´1 x1 q \ G8 pγ´1 x2 q.

Note that Bβ “ tx2 u encodes β in the sense that x2 shadows β. This can be seen
directly as follows: set x02 “ x2 and let xi`1
be the f -lift of xi2 that starts at xi2 p1q.
2
i
Then the sequence of endpoints x2 p1q converges to β, and the infinite concatenation
x02 #x12 # ¨ ¨ ¨ is a path from α to β.
Let now T be the clockwise Dehn twist around the simple closed curve t surr is given by
rounding an interval 0 and β as in Figure 1. The action of T on G
pγ

T˚ : γ´1 Ñ γ´1´1

γβ q´1

pγ´1 γβ q´1

, γβ Ñ γβ

, γ0 Ñ γ0 , γ8 Ñ γ8 .

r of g :“ T ˝ f is B
r b BT˚ . Let us show that g is conjugate to
The biset C
2
p
z ´ 1 : pC, t8, ´1, 0, αuq ý. By Theorem 2.19,
ra , G
r a q r “ pB
ra , G
ra q r EpT q.
pC
aPA
aPA

Note that we have Ca “ Ba for all a P A, so it remains to compute Cβ .
Let t̄ P π1 pCzt0, ´1u, βq be the simple loop below 0 circling around ´1; then
T “ pushpt̄q. Let ℓβ : r0, 1s Ñ Czt´1, 0u (using notations of Lemma 2.16) be an arc
from α to β slightly below 0 so that γβ may be homotoped to a small neighborhood
´1
´1
of ℓβ . We have Cβ “ Bβ γ´1
; the claim then follows from x2 γ´1
“ x1 and the fact
that x1 shadows α.
More generally, suppose that gr “ pushps̄qfrpushpt̄q for some motions s̄, t̄ P
π1 pCzt0, ´1u, βq of β; then
´1
Cβ “ pℓβ #s̄#ℓ´1
β qBβ pℓβ #t̄#ℓβ q.

The process
´1
pℓβ #s̄#ℓ´1
β qx2 pℓβ #t̄#ℓβ q “ g1 xip1q „ xip1q g1 “ g2 xip2q „ xip2q g2 “ . . .

eventually terminates in either x1 or x2 . In the former case, gr is conjugate to z 2 ´
p t8, 0, ´1, αuq ý and in the latter case r
p t8, 0, ´1, βuq ý.
1 : pC,
g is conjugate to z 2 ´1 : pC,
6.1.2. Marking 1 and

?

2. Consider now

?
p t8, ´1, 0, 1, 2uq ý
frpzq “ z 2 ´ 1 : pC,

50

LAURENT BARTHOLDI AND DZMITRY DUDKO

´1
‚

ℓ´1

α
‚

ℓ0

0
‚

´1 1
´α ℓ0 Òf

ℓ1

1
‚?
2
ℓ´1
1 Òf

?
2
‚

ℓ?2

Figure 2. Curves ℓ´1 , ℓ0 , ℓ1 , ℓ?2 and their lifts.
?
?
r “ t1, 2u and A “ t8, ´1, 0u. The dynamics are 2 ÞÑ 1 ÞÑ 0 Ø ´1.
with AzA
r “ Bpfrq in the basis S:
As in §6.1.1 we readily compute a presentation of B
γ´1 “!1, γ0 "p1, 2q,

γ?2 “!1, 1",
γ1 “!1, γ?2 ",

γ0 “!γ´1 , γ1 ",

´1 ´1
´1
"p1, 2q,
γ8 “!γ´1
γ0 , γ1´1 γ?
2

r “ xγ8 , γ´1 , γ? , γ1 , γ0 | γ8 γ´1 γ? γ1 γ0 y.
with G
2
2
As in Lemma 2.16, let ℓ´1 and ℓ0 Ă R respectively be simple arcs from α to ´1
and to 0 so that γ´1 and γ0 may be homotoped to small neighborhoods
of γ´1 and
?
γ0 respectively. Let ℓ1 and ℓ?2 be simple arcs from α to 1 and 2 slightly below 0
as in Figure 2.
r “ Bpfrq. Then
Denote by pGa , Ba qaPAr the portrait of bisets in B induced by B
Ga and Ba are the same as in §6.1.1 for all a P A, while (by Lemma 2.18)
?

1
2
´1
?
?
B1 “ tℓ1 #ℓ´1
0 Òf u “ tx2 u and B 2 “ tℓ 2 #ℓ1 Òf u “ tx2 u.

We consider again some twists of fr. ?Consider gr “ m1 frm2 with m1 , m2 trivial
rel A. Suppose that m1 moves 1 and 2 along s1 P π1 pCzt´1, 0u, 1q and s?2 P
?
?
while m2 moves 1 and 2 along t1 P π1 pCzt´1, 0u, 1q
π1 pCzt´1, 0u, 2q respectively
?
and t?2 P π1 pCzt´1, 0u, 2q respectively. If pGa , Ca qaPAr is the portrait of bisets in
B induced by Bpr
g q, then
(
C1 “ pℓ1 #s1 #ℓ´1
1 qx2 ,
)
!
?
2
´1
?
?
C?2 “ pℓ?2 #s?2 #ℓ´1
1 qx2 pℓ 2 #t1 Òf #ℓ 2 q .

Write C1 “ th1 x2 u and C?2 “ th2 xj u. We may conjugate pC1 , C2 q to ptx2 u, txj h1 uq,
?
p t8, ´1, 0, 1, 2uq ý;
and write xj h1 “ h2 xk . If k “ 1, then r
g is conjugate to z 2 ´ 1 : pC,
?
p t8, ´1, 0, 1, ´ 2uq ý.
otherwise gr is conjugate to z 2 ´ 1 : pC,

6.1.3. Mapping class bisets. We continue the discussion from §6.1.2, and consider
the related mapping class bisets. The biset M pBq is of course reduced to tBu, with
ModpGq “ 1.
r is a left-free ModpGq-biset
r
On the other hand, the biset M pBq
of degree 2.
?
This can be seen in various ways: analytically, the point 2 may be moved to

ERASING MAPS, ORBISPACES, AND THE BIRMAN EXACT SEQUENCE

51

?
?
2
r
1, 2uq, pz 2?´
´ 2, and a basis
? of M pBq may be chosen as tpz ´ 1, t8, ´1, 0, ?
1, t8, ´1, 0, 1, ´ 2uqu. More symbolically, the action of exchanging 2 with ´ 2
amounts to changing, in the wreath recursion of fr, the entry ‘γ1 “ !1, γ?2 "’ into
‘γ1 “ !γ?2 , 1"’.
r is nevertheless connected, and M pBq
r “ M ˚ pBq:
r inNote that the biset M pBq
deed the right action by the mapping class that pushes 1 once along the circle
t|z| “ 1u has the effect of exchanging the two left orbits.
?
Example 6.1. Let us now consider the sets A “ t8, ´1, 0, 1u and D “ t 2u, still
p Aq ý and fr “ f : pC,
p A \ Dq ý. Then the fibres
with the map f pzq “ z 2 ´ 1 : pC,
˚ r
of the map FD,D : M pf q “ M pfrq Ñ M pf q are not connected (by the second claim
of Proposition 2.21).
p
Indeed the left action of ModpCzA,
Dq has two orbits, while the right action
does not identify these orbits since 1 is not allowed to move around a critical value.

6.2. Belyi maps. The Basilica map f pzq “ z 2 ´ 1 is an example of dynamical
Belyi map, namely a map whose post-critical set consists of 3 points. All such
maps are realizable as holomorphic maps, and the three points may be normalized
as t0, 1, 8u.
In this subsection, we briefly state how the main results of this article simplify considerably. We concentrate on the dynamical situation, namely a map
f : pS 2 , Aq ý covered by fr: pS 2 , A \ Dq ý, with #A “ 3.
In that case, ModpS 2 , Aq “ 1, and we have a short exact sequence

1 Ñ knBraidpS 2 , A \ Dq Ñ ModpS 2 , A \ Dq Ñ π1 pS 2 zA, Dq Ñ 1.
Assume first that D consists only of fr-fixed points. Then the extension of bisets
decomposition of M pfrq from Theorem B reduces to the statement that M pfrq is
an inert extension of Bpf qD . The case of D consisting of a single fixed point was
considered in [5, §8.2].
More concretely: if we are given a wreath recursion G Ñ G ≀ S Ó. for Bpf q, with
G “ π1 pS 2 zA, ˚q, then a generating set for ModpS 2 , A\Dq can be chosen to consist
of #D copies of a generating set of G, corresponding to point pushes of D in S 2 zA,
together with some additional knitting elements. A wreath recursion for M pfrq will
then be of the form ModpS 2 , A \ Dq Ñ ModpS 2 , A \ Dq ≀ pS D qÓ., consisting of D
parallel copies of the wreath recursion of Bpf q. The wreath recursion associated
with the knitting elements is trivial.
In case D consists of periodic points of period ą 1, then the actions of G on
the left and the right should be appropriately permuted. If D contains ni cycles
of period i, then abstractly M pfrq will consist in the direct product of ni copies of
Bpf i q “ Bpf qbi .
Example 6.2. Consider the map frpzq “ f pzq “ z 3 with A “ t0, 1, 8u and D “
tω “ expp2πi{3qu. The biset M pf q is of course reduced to tf u, while M pfrq “
p
p
ModpCzA,
Dq – π1 pCzAq
with trivial right action. Indeed D is not in the image
of f , so pushing ω has no effect.
p
However, M ˚ pfrq has two left ModpCzA,
Dq-orbits by Lemma 2.20. Representa3
tives may be chosen as tpz , A\ tωuq, pz 3 , A\ tω 2 uqu, or equivalently (if the marked
p Aq ý
set is to remain A \ D) as the maps tz 3 , z 3 ˝ mu for a homeomorphism m : pC,
2
that pushes ω to ω .

52

LAURENT BARTHOLDI AND DZMITRY DUDKO

6.3. Cyclic bisets. We finally consider the easy case of a cyclic biset, namely the
p t0, 8uq ý for some d P Zzt´1, 0, 1u.
biset G BG of a monomial map f pzq “ z d : pC,
Then G – Z and B is a left-free right-principal biset. Choosing b0 P B we can
identify B “ b0 G with Z with the actions are given by
m ¨ b ¨ n “ dm ` b ` n.

Observe first that b Ñ b ` k is an automorphism of B for all k P Z. Therefore,
contrarily Proposition A we have AutpBq – Z.
Suppose that pGa , Ba qaPAr is a portrait of bisets in B. Then AutpBq acts on
portraits by
pGa , Ba qaPAr ` k :“ pGa , Ba ` kqaPAr .
We denote by pGa , Ba qaPAr { AutpBq the orbit of this action. We can now adjust
Theorem C for cyclic bisets:
r ։ G be a forgetful morphism of groups with G – Z, and let
Lemma 6.3. Let G
p t0, 8uq ý.
B
be
the
biset
of
z d : pC,
G G
There is then a bijection between, on the one hand, conjugacy classes of portraits
of bisets pBa qaPAr in B considered up to the action of AutpBq and, on the other
r G-bisets
r
r ։ G considered up to composition with
hand, Gprojecting to B under G
the biset of a knitting element. This bijection maps every minimal portrait of bisets
r to pBa q r { AutpBq.
of B
aPA

p t0, 8uq so as to remove automorphisms,
Proof. Mark an extra fixed point in pC,
and apply Theorem C.


p t0, 8uYDq ý and let pGa , Ba q r
To illustrate Lemma 6.3, consider frpzq “ z d : pC,
aPA
p t0, 8uq. Let us also assume that d ą 1. For
be the induced portrait of bisets on pC,
a P D write Ba “ tba u with ba P B – Z.
Let us consider a twisted map gr “ mfrn. For a P D let ma and na be the number
of times m and n push a around 0. If pGa , Ca qaPAr is the induced portrait of bisets,
then Ca “ tdma ` bd ` nf˚ paq u “ tca u for a P A. For a P D set
xa – ca {d ` cf˚ paq {d2 ` cf˚2 paq {d3 ` . . .

mod Z P R{Z.

Then pCa qaPD shadows pxa qaPD . The map r
g is unobstructed if and only if all points
xa are pairwise different. If gr is unobstructed, then pxa qdPD considered up to the
action
pxa qaPD Ñ pxa ` kqaPD , k P Z{pd ´ 1qZ
is a complete conjugacy invariant of gr.
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E-mail address: laurent.bartholdi@gmail.com

54

LAURENT BARTHOLDI AND DZMITRY DUDKO

École Normale Supérieure, Paris and Mathematisches Institut, Georg-August Universität zu Göttingen
E-mail address: dzmitry.dudko@gmail.com
Jacobs University, Bremen

