SYMMETRIC AUTOMORPHISMS OF FREE GROUPS,
BNSR-INVARIANTS, AND FINITENESS PROPERTIES

arXiv:1607.03043v1 [] 11 Jul 2016

MATTHEW C. B. ZAREMSKY
Abstract. The BNSR-invariants of a group G are a sequence Σ1 (G) ⊇ Σ2 (G) ⊇ · · ·
of geometric invariants that reveal important information about finiteness properties of
certain subgroups of G. We consider the symmetric automorphism group ΣAutn and
pure symmetric automorphism group PΣAutn of the free group Fn , and inspect their
BNSR-invariants. We prove that for n ≥ 2, all the “positive” and “negative” character
classes of PΣAutn lie in Σn−2 (PΣAutn ) \ Σn−1 (PΣAutn ). We use this to prove that for
n ≥ 2, Σn−2 (ΣAutn ) equals the full character sphere S 0 of ΣAutn but Σn−1 (ΣAutn ) is
empty, so in particular the commutator subgroup ΣAut′n is of type Fn−2 but not Fn−1 .
Our techniques involve applying Morse theory to the complex of symmetric marked cactus
graphs.

Introduction
The Bieri–Neumann–Strebel–Renz (BNSR) invariants Σm (G) of a group G are a sequence of geometric invariants Σ1 (G) ⊇ Σ2 (G) ⊇ · · · that encode a large amount of
information about the subgroups of G containing the commutator subgroup G′ . For example if G is of type Fn and m ≤ n then Σm (G) reveals precisely which such subgroups
are of type Fm . Recall that a group is of type Fn if it admits a classifying space with
compact n-skeleton; these finiteness properties are an important class of quasi-isometry
invariants of groups. The BNSR-invariants are in general very difficult to compute; a complete description is known for the class of right-angled Artin groups [MMV98, BG99], but
not many other substantial families of groups. A complete picture also exists for the generalized Thompson groups Fn,∞ [BGK10, Koc12, Zar15], and the first invariant Σ1 is also
known for some additional classes of groups, e.g., one-relator groups [Bro87], pure braid
groups [KMM15] and pure symmetric automorphism groups of right-angled Artin groups
[OK00, KP14], among others.
In this paper, we focus on the groups ΣAutn and PΣAutn of symmetric and pure symmetric automorphisms of the free group Fn . An automorphism of Fn is symmetric if it
takes each basis element to a conjugate of a basis element, and pure symmetric if it takes
each basis element to a conjugate of itself. These are also known as the (pure) loop braid
groups, and are the groups of motions of n unknotted unlinked oriented loops in 3-space;
Date: July 12, 2016.
2010 Mathematics Subject Classification. Primary 20F65; Secondary 20F28, 57M07.
Key words and phrases. Symmetric automorphism, BNSR-invariant, finiteness properties.

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MATTHEW C. B. ZAREMSKY

an element describes these loops moving around and through each other, ending up back
where they started, either individually in the pure case or just as a set in the non-pure
case (but preserving orientation in both cases). Other names for these and closely related
groups include welded braid groups, permutation-conjugacy automorphism groups, braidpermutation groups and more. See [Dam16] for a discussion of the many guises of these
groups. Some topological properties known for PΣAutn include that is has cohomological
dimension n − 1 [Col89], it is a duality group [BMMM01] and its cohomology ring has been
computed [JMM06].
The first invariant Σ1 (PΣAutn ) was fully computed by Orlandi–Korner [OK00] (she
denotes the group by P Σn ). Koban and Piggott subsequently computed Σ1 (G) for G the
group of pure symmetric automorphisms of any right-angled Artin group [KP14]. One
reason that the question of BNSR-invariants is interesting for PΣAutn is that PΣAutn is
similar to a right-angled Artin group, for instance it admits a presentation in which the
relations are all commutators (see Section 2), but for n ≥ 3 it is not a right-angled Artin
group [KP14], and it is not known whether it is a CAT(0) group [BMMM01, Question 6.4].
The BNSR-invariants are completely known for right-angled Artin groups, but the Morse
theoretic proof of this fact in [BG99] made essential use of the CAT(0) geometry of the
relevant complexes.
Our approach here is to use Morse theory applied to the complex of symmetric marked
cactus graphs ΣKn to prove the following main results:
Theorem A. For n ≥ 2, if χ is a positive or negative1 character of PΣAutn then [χ] ∈
Σn−2 (PΣAutn ) \ Σn−1 (PΣAutn ).
Theorem B. For n ≥ 2, we have Σn−2 (ΣAutn ) = S(ΣAutn ) = S 0 and Σn−1 (ΣAutn ) = ∅.
In particular the commutator subgroup ΣAut′n is of type Fn−2 but not Fn−1 .
For example this shows that ΣAut′n is finitely generated if and only if n ≥ 3, and finitely
presentable if and only if n ≥ 4. It appears that these are already new results (except for
the fact that ΣAut′2 is not finitely generated, which is easy to see since ΣAut2 ∼
= F2 ⋊ S2 ).
Theorem B also provides what could be viewed as the first examples for m ≥ 2 of “naturally
occurring” groups G of type F∞ such that Σm−1 (G) = S(G) but Σm (G) = ∅, and of groups
of type F∞ whose commutator subgroups have arbitrary finiteness properties. (One can
also construct more ad hoc examples: we have noticed that taking a semidirect product of
F2n with the Coxeter group of type Bn = Cn also produces a group with these properties.)
As a remark, contrasting the loop braid group ΣAutn with the classical braid group Bn , it
is easy to see that Σm (Bn ) = S(Bn ) = S 0 for all m and n, and Bn′ is of type F∞ for all n.
For the case n = 3 we can actually get a full computation of Σm (PΣAut3 ); Orlandi–
Korner already computed Σ1 (PΣAut3 ) (it is dense in S(PΣAut3 ); see Citation 2.2), and
we prove that Σ2 (PΣAut3 ) = ∅ (see Theorem 4.23). We tentatively conjecture that
Σn−2 (PΣAutn ) is always dense in S(PΣAutn ) and Σn−1 (PΣAutn ) is always empty, but for
n ≥ 4 it seems this cannot be proved using our techniques, as discussed in Remark 4.24.
1For

the definitions of “positive” and “negative” consult Definition 2.4.

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3

As a remark, there is a result of Pettet involving finiteness properties of some other
normal subgroups of PΣAutn . Namely she found that the kernel of the natural projection
PΣAutn → PΣAutn−1 is finitely generated but not finitely presentable when n ≥ 3 [Pet10].
This is in contrast to the pure braid situation, where the kernel of the “forget a strand”
map P Bn → P Bn−1 is of type F∞ (in fact it is the free group Fn−1 ).
This paper is organized as follows. In Section 1 we recall the background on BNSRinvariants and Morse theory. In Section 2 we discuss the groups of interest, and in Section 3
we discuss the complex ΣKn . We prove Theorem A in Section 4, and with Theorem A in
hand we quickly prove Theorem B in Section 5.
Acknowledgments. I am grateful to Alex Schaefer for a helpful conversation about graph
theory that in particular helped me figure out how to prove that Σ2 (PΣAut3 ) = ∅ (see
Subsection 4.3), to Celeste Damiani for pointing me toward the paper [Sav96], and to
Robert Bieri for enlightening discussions about the novelty of the behavior of the BNSRinvariants found in Theorem B.
1. BNSR-invariants and Morse theory
In this rather technical section we recall the definition of the BNSR-invariants, and set
up the Morse theoretic approach that we will use. The results in Subsection 1.3 are general
enough that we expect they should be useful in the future to compute BNSR-invariants of
other interesting groups.
1.1. BNSR-invariants. A CW-complex Z is called a classifying space for G, or K(G, 1),
if π1 (Z) ∼
= G and πk (Z) = 0 for all k 6= 1. We say that G is of type Fn if it admits a
K(G, 1) with compact n-skeleton. For example G is of type F1 if and only if it is finitely
generated and of type F2 if and only if it is finitely presentable. If G is of type Fn for all
n we say it is of type F∞ . If G acts properly and cocompactly on an (n − 1)-connected
CW-complex, then G is of type Fn .
Definition 1.1 (BNSR-invariants). Let G be a group acting properly and cocompactly on
an (n − 1)-connected CW-complex Y (so G is of type Fn ). Let χ : G → R be a character
of G, i.e., a homomorphism to R. There exists a map hχ : Y → R, which we will call a
character height function, such that hχ (g.y) = χ(g) + hχ (y) for all g ∈ G and y ∈ Y . For
t ∈ R let Yχ≥t be the full subcomplex of Y supported on those 0-cells y with hχ (y) ≥ t.
Let [χ] be the equivalence class of χ under scaling by positive real numbers. The character
sphere S(G) is the set of non-trivial character classes [χ]. For m ≤ n, the mth BNSRinvariant Σm (G) is defined to be
Σm (G) := {[χ] ∈ S(G) | (Yχ≥t )t∈R is essentially (m − 1)-connected}.
Recall that (Yχ≥t )t∈R is said to be essentially (m − 1)-connected if for all t ∈ R there
exists −∞ < s ≤ t such that the inclusion of Yχ≥t into Yχ≥s induces the trivial map in πk
for all k ≤ m − 1.

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MATTHEW C. B. ZAREMSKY

It turns out Σm (G) is well defined up to the choice of Y and hχ (see for example [Bux04,
Definition 8.1]). As a remark, the definition there used the filtration by sets h−1
χ ([t, ∞))t∈R ,
but thanks to cocompactness this filtration is essentially (m − 1)-connected if and only if
our filtration (Yχ≥t )t∈R is.
One important application of BNSR-invariants is the following:
Citation 1.2. [BR88, Theorem B and Remark 6.5] Let G be a group of type Fm . Let
G′ ≤ H ≤ G. Then H is of type Fm if and only if for every non-trivial character χ of G
such that χ(H) = 0, we have [χ] ∈ Σm (G).
For example, if H = ker(χ) for χ a discrete character of G, i.e., one with image Z, then
H is of type Fm if and only if [±χ] ∈ Σm (G). Also note that G′ itself is of type Fm if and
only if Σm (G) = S(G).
Other important classical properties of the Σm (G) are that they are all open subsets of
S(G) and that they are invariant under the natural action of Aut(G) on S(G) [BNS87,
BR88].
1.2. Morse theory. Bestvina–Brady Morse theory can be a useful tool for computing
BNSR-invariants. In this section we give the relevant definitions and results from Morse
theory, in the current level of generality needed.
Let Y be an affine cell complex (see [BB97, Definition 2.1]). The star stY v of a 0-cell v
in Y is the subcomplex of Y consisting of cells that are faces of cells containing v. The link
lkY v of v is the simplicial complex stY v of directions out of v into stY v. We will suppress
the subscript Y from the notation when it is clear from context. If v and w are distinct
0-cells sharing a 1-cell we will call v and w adjacent and write v adj w.
In [BB97], Bestvina and Brady defined a Morse function on an affine cell complex Y to
be a map Y → R that is affine on cells, takes discretely many values on the 0-cells, and is
non-constant on 1-cells. When using Morse theory to compute BNSR-invariants though,
these last two conditions are often too restrictive. The definition of Morse function that
will prove useful for our purposes is as follows.
Definition 1.3 (Morse function). Let Y be an affine cell complex and let h : Y → R and
f : Y → R be functions that are affine on cells. We call (h, f ) : Y → R × R a Morse
function if the set {h(v) − h(w) | v, w ∈ Y (0) , v adj w} does not have 0 as a limit point
(we will call it discrete near 0), the set {f (v) | v ∈ Y (0) } is finite2, and if v, w ∈ Y (0) with
v adj w and h(v) = h(w) then f (v) 6= f (w).
For example if h takes discrete values on 0-cells and distinct values on adjacent 0-cells,
then (taking f to be constant and ignoring it) we recover Bestvina and Brady’s notion of
“Morse function”.
2In

what follows it will be clear that “finite” could be replaced with “well ordered” but for our present
purposes we will just assume it is finite.

SYMMETRIC AUTOMORPHISMS, BNSR-INVARIANTS, AND FINITENESS PROPERTIES

5

Using the usual order on R and the lexicographic order on R × R, it makes sense to
compare (h, f ) values of 0-cells. On a given cell c, since h and f are affine on c it is clear
that (h, f ) achieves its maximum and minimum values at unique faces of c, and the last
assumption in Definition 1.3 ensures these will be 0-cells.
Definition 1.4 (Ascending star/link). Given a Morse function (h, f ) on an affine cell
complex Y , define the ascending star st↑ v of a 0-cell v in Y to be the subcomplex of st v
consisting of all faces of those cells c for which the unique 0-face of c where (h, f ) achieves
its minimum is v. Define the ascending link lk↑ v to be the subcomplex of lk v consisting
of directions into st↑ v. Note that lk↑ v is a full subcomplex of lk v, since h and f are affine
on cells.
For Y an affine cell complex, (h, f ) a Morse function on Y and t ∈ R, denote by Yh≥t
the subcomplex of Y supported on those 0-cells v with h(v) ≥ t.
Lemma 1.5 (Morse Lemma). Let Y be an affine cell complex and (h, f ) : Y → R × R
a Morse function. Let t ∈ R and s ∈ [−∞, t). If for all 0-cells v with h(v) ∈ [s, t)
the ascending link lk↑ v is (m − 1)-connected, then the inclusion Yh≥t → Yh≥s induces an
isomorphism in πk for k ≤ m − 1 and an epimorphism in πm .
Proof. The essential parts of the proof are the same as in [BB97]. Choose ε > 0 such that
for any v adj w, |h(v) − h(w)| 6∈ (0, ε) (this is possible since the set of values h(v) − h(w) for
v adj w is discrete near 0). We can assume by induction (and by compactness of spheres if
s = −∞) that t−s ≤ ε. In particular if adjacent 0-cells v and w both lie in Yh≥s \Yh≥t , then
h(v) = h(w) and f (v) 6= f (w). To build up from Yh≥t to Yh≥s , we need to glue in the 0-cells
of Yh≥s \ Yh≥t along their relative links in some order such that upon gluing in v, all of lk↑ v
is already present, but nothing else in lk v, so the relative link is precisely the ascending
(0)
(0)
link. To do this, we put any order we like on each set Fi := {v ∈ Yh≥s \ Yh≥t | f (v) = i} for
(0)
(0)
i ∈ f (Y (0) ), and then extend these to an order on Yh≥s \ Yh≥t by declaring that everything
in Fi comes after everything in Fj whenever i < j. Now when we glue in v, for w ∈ lk v we
have w ∈ lk↑ v if and only if either h(w) > h(v), in which case h(w) ≥ t and w is already
present, or h(w) = h(v) and f (w) > f (v), in which case w ∈ Ff (w) is also already present.
Since the relevant ascending links are (m − 1)-connected by assumption, the result follows
from the Seifert–van Kampen, Mayer–Vietoris and Hurewicz Theorems.

As a corollary to the proof, we have:
Corollary 1.6. With the same setup as the Morse Lemma, if additionally for all 0-cells
e m+1 (lk↑ v) = 0, then the inclusion Yh≥t → Yh≥s induces an
v with s ≤ h(v) < t we have H
e m+1 .
injection in H
Proof. In the proof of the Morse Lemma, we saw that Yh≥s is obtained from Yh≥t by coning
off the ascending links of 0-cells v with s ≤ h(v) < t, so this is immediate from the
Mayer–Vietoris sequence.


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MATTHEW C. B. ZAREMSKY

For example if Y is (m + 1)-dimensional, so the links are at most m-dimensional, then
this additional condition will always be satisfied.
Wen dealing with BNSR-invariants, the following is particularly useful:
Corollary 1.7. Let Y be an (m − 1)-connected affine cell complex with a Morse function
(h, f ). Suppose there exists q such that, for every 0-cell v of Y with h(v) < q, lk↑ v is (m −
1)-connected. Then the filtration (Yh≥t )t∈R is essentially (m − 1)-connected. Now assume
e m+1 (Y ) = 0 and for every 0-cell v of Y with h(v) < q, H
e m+1 (lk↑ v) = 0,
additionally that H
e m (lk↑ v) 6= 0. Then the
and that for all p there exists a 0-cell v with h(v) < p such that H
filtration (Yh≥t)t∈R is not essentially m-connected.
Proof. By the Morse Lemma, for any r ≤ q the inclusion Yh≥r → Y = Yh≥−∞ induces an
isomorphism in πk for k ≤ m − 1. Since Y is (m − 1)-connected, so is Yh≥r . Now for any
t ∈ R we just need to choose s = min{q, t} and we get that the inclusion Yh≥t → Yh≥s
induces the trivial map in πk for k ≤ m − 1, simply because Yh≥s is (m − 1)-connected.
For the second claim, suppose that (Yh≥t )t∈R is essentially m-connected. Say t < q, and
choose s ≤ t such that the inclusion Yh≥t → Yh≥s induces the trivial map in πk for k ≤ m.
Also, since t < q, this inclusion induces a surjection in these πk by the Morse Lemma, so in
fact Yh≥s itself is m-connected, as are all Yh≥r for r ≤ s (for the same reason). Now choose
e m (lk↑ v) 6= 0. Since H
e m (Yh≥r ) = 0 for all r ≤ s, Mayer–Vietoris
v such that h(v) < s and H
e m+1 (Yh≥q ) 6= 0 for any q ≤ h(v). But this includes q = −∞,
and Corollary 1.6 say that H
e m+1 (Y ) = 0.
which contradicts our assumption that H

1.3. BNSR-invariants via Morse theory. We now return to the situation in Definition 1.1, so Y is an (n − 1)-connected CW-complex on which G acts properly and cocompactly (and, we assume, cellularly), χ is a character of G, and hχ is a character height
function on Y . The goal of this subsection is to establish a Morse function on Y using hχ .
Let us make two additional assumptions. First, assume Y is simplicial (this is just to
ensure that any function on Y (0) can be extended to a function on Y that is affine on cells).
Second, assume that no adjacent 0-simplicies in Y share a G-orbit (if this is not the case,
it can be achieved by subdividing). Let f : Y (0) /G → R be any function that takes distinct
values on adjacent 0-cells, where the cell structure on Y /G is induced from Y . (Just to give
some examples, one could construct f by randomly assigning distinct values to the 0-cells
in Y /G, or one could take the barycentric subdivision and have f read the dimension.)
Define f : Y (0) → R via f (v) := f (G.v), and extend f to a map (also called f ) on all of Y
by extending affinely to each simplex.
Lemma 1.8. With Y , hχ and f as above, (hχ , f ) : Y → R × R is a Morse function.
Proof. The functions hχ and f are affine on cells by construction. The set {f (v) | v ∈ Y (0) }
equals the set {f (G.v) | G.v ∈ Y (0) /G}, which is finite since Y /G is compact. For any
g ∈ G we have hχ (g.v) − hχ (g.w) = χ(g) + hχ (v) − χ(g) − hχ (w) = hχ (v) − hχ (w), so

SYMMETRIC AUTOMORPHISMS, BNSR-INVARIANTS, AND FINITENESS PROPERTIES

7

by compactness of Y /G the set {hχ (v) − hχ (w) | v, w ∈ Y (0) , v adj w} is finite (and hence
discrete near 0). Finally, since f takes distinct values on adjacent 0-cells in Y /G, and no
adjacent 0-cells in Y share an orbit, we see f takes distinct values on adjacent 0-cells in
Y.

In particular Corollary 1.7 can now potentially be used to prove that (Yχ≥t )t∈R is or is
not essentially (m − 1)-connected, and hence that [χ] is or is not in Σm (G).
While any f constructed as above will make (hχ , f ) a Morse function, this does not mean
every f may be useful, for instance if the ascending links are not as highly connected as
one would hope. In fact it seems likely that situations exist where every choice of f yields
a “useless” Morse function. Hence, in practice one hopes to have a concrete space Y with
a natural choice of f that produces nice ascending links.
2. (Pure) symmetric automorphism groups
We now turn to our groups of interest.
Let Fn be the free group with basis S := {x1 , . . . , xn }. An automorphism α ∈ Aut(Fn )
is called symmetric if for each i ∈ [n] := {1, . . . , n}, (xi )α is conjugate to xj for some
j; if each (xi )α is conjugate to xi we call α pure symmetric 3. Note that in some texts,
“symmetric” allows for (xi )α to be conjugate to some x−1
j , but we do not allow that here.
Denote by ΣAutn the group of all symmetric automorphisms of Fn , and by PΣAutn the
group of pure symmetric automorphisms. The abelianization Fn → Zn induces a surjection
Aut(Fn ) → GLn (Z), and the restriction of this map to ΣAutn yields a splitting
ΣAutn ∼
= PΣAutn ⋊Sn .
An equivalent description of ΣAutn (and PΣAutn ) is as the (pure) loop braid group,
i.e., the group of (pure) motions of n unknotted, unlinked oriented circles in 3-space.
The subgroup of ΣAutn consisting of those automorphisms taking x1 · · · xn to itself is
isomorphic to the classical braid group Bn , and the intersection of this with PΣAutn is
the classical pure braid group P Bn [Sav96]. Other names for ΣAutn and closely related
groups include welded braid groups, permutation-conjugacy automorphism groups, braidpermutation groups and more. Details on the various viewpoints for these groups can be
found for example in [Dam16].
In [McC86], McCool found a (finite) presentation for PΣAutn . The generators are the
automorphisms αi,j (i 6= j) given by (xi )αi,j = x−1
j xi xj and (xk )αi,j = xk for k 6= i, and
the defining relations are [αi,j , αk,ℓ ] = 1, [αi,j , αk,j ] = 1 and [αi,j αk,j , αi,k ] = 1, for distinct
i, j, k, ℓ. (In particular note that this implies PΣAut2 ∼
= F2 .) It will also be convenient
later to consider automorphisms αI,j , defined via
Y
αI,j :=
αi,j
i∈I
3Automorphisms

will be acting on the right here, so we will reflect this in the notation.

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MATTHEW C. B. ZAREMSKY

for I ⊆ [n] \ {j}, where the product can be taken in any order thanks to the relation
[αi,j , αk,j ] = 1. Following Collins [Col89] we call these symmetric Whitehead automorphisms.
Since the defining relations in McCool’s presentation are commutators, we immediately
see that PΣAutn has abelianization Zn(n−1) , with basis {αi,j | i 6= j}. Since Sn acts
transitively on the αi,j , we also quickly compute that ΣAutn abelianizes to Z × (Z/2Z)
for all n ≥ 2. A natural basis for the vector space Hom(PΣAutn , R) ∼
= Rn(n−1) is the dual
of {αi,j | i 6= j}. This dual basis has a nice description that we will now work up to.
w
For α ∈ PΣAutn let wi,α ∈ Fn be the elements such that (xi )α = xi i,α . For each i 6= j
define χi,j : PΣAutn → Z by sending α to ϕj (wi,α), where ϕj : Fn → Z are the projections
sending xj to 1 and the other generators to 0.
Lemma 2.1. Each χi,j is a homomorphism.
Proof. Let α, β ∈ PΣAutn and i ∈ [n]. Write wi,α = xεk11 · · · xεkrr for k1 , . . . , kr ∈ [n] and
ε1 , . . . , εr ∈ {±1}, so we have
−1
1
r
)β · · · (xk−ε
)βwi,β
(xi )α ◦ β = (xk−ε
xi wi,β (xεk11 )β · · · (xεkrr )β.
r
1

In particular
wi,α◦β = wi,β (xεk11 )β · · · (xεkrr )β.
Note that ϕj ((xεk11 )β · · · (xεkrr )β) = ϕj (xεk11 · · · xεkrr ), so ϕj (wi,α◦β ) = ϕj (wi,α ) + ϕj (wi,β ), as
desired.

Clearly χi,j (αk,ℓ ) = δ(i,j),(k,ℓ) (the Kronecker delta), so {χi,j | i 6= j} is the basis of
Hom(PΣAutn , R) dual to {αi,j | i 6= j}. Since Hom(PΣAutn , R) ∼
= Rn(n−1) we know that
n(n−1)−1
the character sphere S(PΣAutn ) is S(PΣAutn ) = S
.
For the group ΣAutn , Hom(ΣAutn , R) ∼
= R for all n, so to find a basis we just need a
non-trivial character. We know ΣAutn = PΣAutn ⋊Sn , so the most natural candidate is
the character reading 1 on each αi,j and 0 on Sn . Note that S(ΣAutn ) = S 0 for all n ≥ 2.
P
Writing an arbitrary character of PΣAutn as χ =
i6=j ai,j χi,j , we recall Orlandi–
1
Korner’s computation of Σ (PΣAutn ):
Citation 2.2. [OK00] We have [χ] ∈ Σ1 (PΣAutn ) unless either
(i) there exist distinct i and j such that ap,q = 0 whenever {p, q} 6⊆ {i, j}, or
(ii) there exist distinct i, j and k such that ap,q = 0 whenever {p, q} 6⊆ {i, j, k} and
moreover ap,q = −ap′ ,q whenever {p, p′ , q} = {i, j, k}.
In these cases [χ] 6∈ Σ1 (PΣAutn ).
For example, Σ1 (PΣAut2 ) is empty (which we know anyway since PΣAut2 ∼
= F2 ) and
Σ (PΣAut3 ) is a 5-sphere with three 1-spheres and one 2-sphere removed, so in particular
Σ1 (PΣAut3 ) is dense in S(PΣAut3 ).
1

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9

The groups ΣAutn and PΣAutn are of type F∞ (this can be seen for example after work
of Collins [Col89])4, so one can ask what Σm (ΣAutn ) and Σm (PΣAutn ) are, for any m and
n. One thing we know, which we will use later, is that the invariants are all closed under
taking antipodes:
Observation 2.3. If [χ] ∈ Σm (G) for G = ΣAutn or PΣAutn then [−χ] ∈ Σm (G).
Proof. The automorphism of Fn taking each xi to x−1
induces an automorphism of ΣAutn
i
and PΣAutn under which each character χ maps to −χ. The result now follows since
Σm (G) is invariant under Aut(G).

We can now state our main results.
Definition 2.4 (Positive/negative character). Call χ =
all i, j, and negative if ai,j < 0 for all i, j.

P

i6=j

ai,j χi,j positive if ai,j > 0 for

Theorem A. For n ≥ 2, if χ is a positive or negative character of PΣAutn then [χ] ∈
Σn−2 (PΣAutn ) \ Σn−1 (PΣAutn ).
As a remark, thanks to Observation 2.3 the negative character classes lie in a given
Σm (PΣAutn ) if and only if the positive ones do, so we only need to prove Theorem A for
positive characters.
Theorem B. For n ≥ 2, we have Σn−2 (ΣAutn ) = S(ΣAutn ) = S 0 and Σn−1 (ΣAutn ) = ∅.
In particular the commutator subgroup ΣAut′n is of type Fn−2 but not Fn−1 .
The commutator subgroups ΣAut′n and PΣAut′n are easy to describe; see for example
Lemmas 4 and 5 of [Sav96]. The commutator subgroup PΣAut′n consists of those automorphisms taking each xi to wi−1xi wi for some wi ∈ Fn′ . In other words, PΣAut′n is just
the intersection of all the ker(χi,j ). Note that for n ≥ 2 the commutator subgroup PΣAut′n
is not finitely generated, since it surjects onto PΣAut′2 ∼
= F2′ . The commutator subgroup
ΣAut′n consists of those automorphisms taking xi to wi−1 xπ(i) wi for some even permutation π ∈ Sn (i.e., π ∈ An ) and satisfying ϕ(w1 · · · wn ) = 0 where ϕ : Fn → Z is the map
taking each basis element to 1. As a remark, the abelianization map ΣAutn → Z splits,
for instance by sending Z to hα1,2 i, so we have ΣAutn = ΣAut′n ⋊Z.
In Section 5 we will be able to deduce Theorem B from Theorem
P A quickly by using the
next lemma. If we write BBn for the kernel of the character i6=j χi,j of PΣAutn taking
each αi,j to 1, so BBn is the “Bestvina–Brady-esque” subgroup of PΣAutn , then we have:
Lemma 2.5. ΣAut′n = BBn ⋊ An .
Proof. When we restrict the map ΣAutn → Sn to ΣAut′n , by the above description we know
that the image is An . This map splits, and the kernel of this restricted map is the kernel of
the original map, which is PΣAutn , intersected with ΣAut′n . The above description tells us
4Actually,

PΣAutn is even of “type F”, meaning it has a compact classifying space, but we will not
need this fact.

10

MATTHEW C. B. ZAREMSKY

that this consists of all pure symmetric automorphisms α such that
Pϕ(w1,α · · · wn,α) = 0,
and from the definition of the χi,j it is clear that ϕ(w1,α · · · wn,α ) = i6=j χi,j (α), so we are
done.

In particular BBn has finite index in ΣAut′n , so they have the same finiteness properties.
To prove Theorems A and B we need a complex on which the groups act nicely, and to
understand ascending links. We discuss the complex in Section 3 and the ascending links
in Section 4.
3. The complex of symmetric marked cactus graphs
In [Col89], Collins found a contractible simplicial complex ΣKn on which the “Outer”
versions of ΣAutn and PΣAutn act properly and cocompactly, described by symmetric
marked cactus graphs. We will use the obvious analog of this complex for our groups.
Thanks to the action being proper and cocompact, and the complex being contractible, it
can be used to “reveal” the BNSR-invariants of the groups, as per Definition 1.1. In this
section we recall the construction of ΣKn and set up the character height functions that
will then be used in the following sections to prove our main results.
Terminology: By a graph we will always mean a connected finite directed graph with
one vertex specified as the basepoint, such that the basepoint has degree at least two and
all other vertices have degree at least three. We will use the usual terminology of initial
and terminal endpoints of an edge, paths, cycles, reduced paths, simple cycles, subtrees,
subforests and spanning trees.
Our graphs will always be understood to have rank n, unless otherwise specified.
Let Rn be the n-petaled rose, that is the graph with one vertex ∗ (which is necessarily
the basepoint) and n edges. Then π1 (Rn ) ∼
= Fn , and we identify Aut(Fn ) with the group
of basepoint-preserving self-homotopy equivalences of Rn , modulo homotopy.
Definition 3.1 (Cactus graph, cladode, base, above, projection, before/after, between).
A graph Γ is called a cactus graph if every edge is contained in precisely one simple cycle.
We will refer to the simple cycles, viewed as subgraphs, as cladodes. For example the petals
of the rose Rn are precisely its cladodes. We will assume the orientations of the edges are
such that each cladode is a directed cycle, that is, no distinct edges of a cladode share an
origin (or terminus). If C is a cladode of a cactus graph Γ with basepoint p, there is a
unique vertex bC of C closest to p, which we call the base of C. Note that every vertex is
the base of at least one cladode. We say that a cladode C is above a cladode D if every
path from bC to p must pass through an edge of D. If C is above D there is a unique vertex
projD (C) of D closest to bC , which we will call the projection of C onto D. Given two
distinct points x and y in a common cladode C, with x, y 6= bC , there is a unique reduced
path from x to y in C \ {bC }; if this path follows the orientation of C we say x is before
y, and otherwise we say x is after y. Within C it also makes sense to say that an edge is
before or after another edge, or that an edge is before or after a point not in the interior

SYMMETRIC AUTOMORPHISMS, BNSR-INVARIANTS, AND FINITENESS PROPERTIES

11

of that edge. We say a point or edge is between two points or edges if it is before one and
after the other.
See Figure 1 for an example illustrating the many definitions in Definition 3.1.
9
8

10
3

7

4

6

11

2
1
5

12

Figure 1. A cactus graph, with its cladodes numbered for reference. To illustrate the definitions in Definition 3.1 with some examples, we note: cladodes 3 and 4 have the same base; cladode 7 is above cladodes 2 and 1 but no
others; the projection of cladode 8 onto cladode 1 is the vertex that is the
base of cladode 3; and the base of cladode 3 is after the base of cladode 2
and before the base of cladode 12, and hence is between them.
Given a graph Γ and a subforest F , we will write Γ/F for the graph obtained by quotienting each connected component of F to a point. The quotient map d : Γ → Γ/F is
called a forest collapse or forest blow-down. It is a homotopy equivalence, and a homotopy
inverse of a forest blow-down is called a forest blow-up, denoted u : Γ/F → Γ.
Definition 3.2 (Marking). A marking of a basepointed graph Γ is a homotopy equivalence
ρ : Rn → Γ from the n-petaled rose to Γ, taking basepoint to basepoint.
A marking of the rose itself represents an automorphism of Fn , thanks to our identification of Aut(Fn ) with the group of basepoint-preserving self-homotopy equivalences of
Rn , modulo homotopy. If a marking α : Rn → Rn even represents a (pure) symmetric
automorphism, then it makes sense to call the marking itself (pure) symmetric. More
generally:
Definition 3.3 ((Pure) symmetric marking). A marking ρ : Rn → Γ is called (pure) symmetric if there exists a forest collapse d : Γ → Rn such that d ◦ ρ : Rn → Rn is (pure)
symmetric.
Definition 3.4 (Symmetric marked cactus graph). A symmetric marked cactus graph is a
triple (Γ, p, ρ) where Γ is a cactus graph with basepoint p and ρ is a symmetric marking.

12

MATTHEW C. B. ZAREMSKY

Two such triples (Γ, p, ρ), (Γ′ , p′ , ρ′ ) are considered equivalent if there is a homeomorphism
φ : Γ → Γ′ taking p to p′ such that φ◦ρ ≃ ρ′ . We will denote equivalence classes by [Γ, p, ρ],
and will usually just refer to [Γ, p, ρ] as a symmetric marked cactus graph.
We note that under this equivalence relation, every symmetric marked cactus graph is
equivalent to one where the marking is even pure symmetric. This is just because the
markings of the rose that permute the petals are all equivalent to the trivial marking.
Moreover, these are the only markings equivalent to the trivial marking, so the map α 7→
[Rn , ∗, α] is in fact a bijection between PΣAutn and the set of symmetric marked roses.
Definition 3.5 (Partial order). We define a partial order ≤ on the set of symmetric
marked cactus graphs as follows. Let [Γ, p, ρ] be a symmetric marked cactus graph and F
a subforest of Γ, with d : Γ → Γ/F the forest collapse. Let pF := d(p) and let ρF := d ◦ ρ.
We declare that [Γ, p, ρ] ≤ [Γ/F, pF , ρF ]. It is easy to check that the relation ≤ is well
defined up to equivalence of triples, and that it is a partial order.
Definition 3.6 (Complex of symmetric marked cactus graphs). The complex of symmetric
marked cactus graphs ΣKn is the geometric realization of the partially ordered set of
symmetric marked cactus graphs.
(0)

Note that ΣAutn and PΣAutn act (on the right) on ΣKn via [Γ, p, ρ].α := [Γ, p, ρ ◦ α],
and this extends to an action on ΣKn since for any forest collapse d : Γ → Γ/F , we have
(d ◦ ρ) ◦ α = d ◦ (ρ ◦ α), i.e., ρF ◦ α = (ρ ◦ α)F .
Citation 3.7. [Col89, Proposition 3.5, Theorem 4.7] The complex ΣKn is contractible
and (n − 1)-dimensional, and the actions of ΣAutn and PΣAutn on ΣKn are proper and
cocompact.
Technically Collins considers the “Outer” version where we do not keep track of basepoints, but it is straightforward to get these results also in our basepointed “Auter” version.
In particular, we have the requisite setup of Definition 1.1.
Remark 3.8. One can similarly consider the complex of all marked basepointed graphs,
and get the well studied spine of Auter space, which is contractible and on which Aut(Fn )
acts properly and cocompactly (see [CV86, HV98]). This is not relevant for our present
purposes though, since the abelianization of Aut(Fn ) is finite, and hence its character
sphere is empty.
The next step is to take a character χ of PΣAutn and induce a character height function
hχ on ΣKn . First recall that we equivocate between symmetric markings of roses and
elements of PΣAutn . Hence for 0-simplices in ΣKn of the form [Rn , ∗, α], we can just
define hχ ([Rn , ∗, α]) := χ(α). In general we define hχ ([Γ, p, ρ]) as follows:
Definition 3.9 (The character height function hχ ). Let [Γ, p, ρ] be a symmetric marked
cactus graph. Define hχ ([Γ, p, ρ]) to be
hχ ([Γ, p, ρ]) := max{χ(α) | [Γ, p, ρ] ∈ st([Rn , ∗, α])}.

SYMMETRIC AUTOMORPHISMS, BNSR-INVARIANTS, AND FINITENESS PROPERTIES

13

Extend this affinely to the simplices of ΣKn , to get hχ : ΣKn → R.
Observation 3.10. hχ is a character height function.
(0)

Proof. We need to show that hχ ([Γ, p, ρ].α) = hχ ([Γ, p, ρ]) + χ(α) for all [Γ, p, ρ] ∈ ΣKn
and α ∈ PΣAutn . We know that [Γ, p, ρ].α = [Γ, p, ρ◦α], and clearly [Γ, p, ρ] ∈ st[Rn , ∗, β] if
and only if [Γ, p, ρ◦α] ∈ st[Rn , ∗, β◦α], so this follows simply because χ(β◦α) = χ(β)+χ(α)
for all α, β ∈ PΣAutn .

Now we need a “tiebreaker” function f as in Lemma 1.8. As discussed before and
(0)
after that lemma, any randomly chosen injective f : ΣKn /G → R could serve to induce
a tiebreaker f : ΣKn → R, but we want to be more clever than this. In particular our
tiebreaker will yield tractable ascending links that will actually reveal parts of the BNSRinvariants.
The 0-cells in the orbit space ΣKn / PΣAutn are homeomorphism classes of cactus graphs,
(0)
so “number of vertices” is a well defined measurement on these 0-cells. Let f : ΣKn /G →
R be the function taking a graph to the negative of its number of vertices. In particular
since we are using the negative, the rose has the largest f value of all cactus graphs. Let
f : ΣKn → R be the extension of f described before Lemma 1.8, so f ([Γ, p, ρ]) equals
the negative number of vertices of Γ, and consider the function (hχ , f ) : ΣKn → R × R.
Since ΣKn is simplicial and adjacent 0-simplices in ΣKn cannot share a PΣAutn -orbit (for
instance since they necessarily have different f values), the following is immediate from
Lemma 1.8:
Corollary 3.11. For any χ, (hχ , f ) is a Morse function on ΣKn .



h ≥t

It is clear from the definition of hχ that ΣKn χ is the union of the stars of those [Rn , ∗, α]
with χ(α) ≥ t. It is a common phenomenon when working in Auter space and its relatives
to encounter important subcomplexes that are unions of stars of marked roses. Another
example arises in [BBM07], where Bestvina, Bux and Margalit use “homology markings” of
roses to prove that for n ≥ 3 the kernel of Out(Fn ) → GLn (Z) has cohomological dimension
2n − 4 and is not of type F2n−4 (when n ≥ 4 it remains open whether or not this kernel is
of type F2n−5 ).
We record here a useful technical lemma that gives information on how hχ can differ
between “nearby” symmetric marked roses. Let [Γ, p, ρ] be a symmetric marked cactus
graph and let T be a spanning tree in Γ. Since T is spanning, collapsing T yields a
symmetric marked rose. The marking ρ provides the cladodes of Γ with a numbering from
1 to n; let Ci,ρ be the ith cladode. Since T is a spanning tree, it meets Ci,ρ at all but
one edge; write Ei,T for the single-edge subforest of Ci,ρ that is not in T . In particular,
intuitively, upon collapsing T , Ei,T becomes the ith petal of Rn . Note that T is completely
determined by the set {E1,T , . . . , En,T }, namely it consists of all the edges of Γ not in any
Ei,T .

14

MATTHEW C. B. ZAREMSKY

Lemma 3.12 (Change of spanning tree). Let [Γ, p, ρ] be a symmetric marked cactus graph
and let T be a spanning tree in Γ. Suppose U is another spanning tree such that Ej,T 6= Ej,U
but Ei,T = Ei,U for all i 6= j (so U differs from T only in the jth cladode). Suppose that
Ej,T is before Ej,U (in the language of Definition 3.1). Let ∅ =
6 I ( [n] be the set of indices
i such that the projection projCj,ρ (Ci,ρ ) lies between Ej,T and Ej,U (so in particular j 6∈ I).
P
P
Then for any χ = i6=j ai,j χi,j we have hχ ([Γ/T, pT , ρT ]) − hχ ([Γ/U, pU , ρU ]) = i∈I ai,j .
Proof. By collapsing the subforest T ∩ U we can assume without loss of generality that
T = Ej,T and U = Ej,U are each a single edge, so Γ has two vertices, the basepoint p
and another vertex q. The set I indexes those cladodes whose base is q, so [n] \ I indexes
those cladodes whose base is p. Up to the action of PΣAutn we can assume
P that Γ/U
is the trivially marked rose, so we need to show that hχ ([Γ/T, pT , ρT ]) =
i∈I ai,j . In
fact the procedure of blowing up the trivial rose to get Γ and then blowing down T is a
Whitehead move (see [CV86, Section 3.1]) that corresponds to the symmetric Whitehead
automorphism αI,j . In other words, viewed as P
an element of PΣAutn , we have ρT = αI,j .
This means that hχ ([Γ/T, pT , ρT ]) = χ(αI,j ) = i∈I ai,j , as desired.

Ascending links: In the next section we will need to understand ascending links lk↑ v
with respect to (hχ , f ), for v = [Γ, p, ρ] a 0-simplex in ΣKn , so we discuss this a bit here.
Since lk↑ v is a full subcomplex of lk v we just need to understand which 0-simplices of
lk v lie in lk↑ v. First note that lk v is a join, of the down-link lkd v, spanned by those
0-simplices of lk v obtained from forest blow-downs of Γ, and its up-link lku v, spanned by
those 0-simplices of lk v corresponding to forest blow-ups of Γ. The ascending link lk↑ v
similarly decomposes as the join of the ascending down-link lk↑d v and ascending up-link
lk↑u v, which are just defined to be lk↑d v := lkd v ∩ lk↑ v and lk↑u v := lku v ∩ lk↑ v.
Since 0-simplices in lkd v have larger f value than v (i.e., the graphs have fewer vertices than Γ) and cannot have strictly larger hχ value, given a subforest F ⊆ Γ we see
that [Γ/F, pF , ρF ] ∈ lk↑d v if and only if hχ ([Γ/F, pF , ρF ]) ≥ hχ ([Γ, p, ρ]) if and only if
e pe, ρe] ∈ lku v then it lies in lk↑ v if and only
hχ ([Γ/F, pF , ρF ]) = hχ ([Γ, p, ρ]). Similarly if [Γ,
u
if it has strictly larger hχ value than v (since it has smaller f value).
4. Topology of ascending links
Throughout this section, χ is a non-trivial character of PΣAutn with character height
function hχ on ΣKn , and lk↑ [Γ, p, ρ] means the ascending link of the 0-simplex [Γ, p, ρ] with
respect to χ.
We will analyze the topology of lk↑ [Γ, p, ρ] = lk↑d [Γ, p, ρ] ∗ lk↑u [Γ, p, ρ] by inspecting
lk↑d [Γ, p, ρ] and lk↑u [Γ, p, ρ] individually. First we focus on lk↑d [Γ, p, ρ].
4.1. Ascending down-link. The first goal is to realize lk↑d [Γ, p, ρ] as a nice combinatorial
object, the complex of ascending forests.

SYMMETRIC AUTOMORPHISMS, BNSR-INVARIANTS, AND FINITENESS PROPERTIES

15

Definition 4.1 (Complex of forests). The complex of forests F (Γ) for a graph Γ is the
geometric realization of the partially ordered set of non-empty subforests of Γ, with partial
order given by inclusion.
We will not really need to know the homotopy type of F (Γ) in what follows, but it is
easy to compute so we record it here for good measure.
Observation 4.2. For Γ a cactus graph with V vertices, F (Γ) ≃ S V −2 .
Proof. For each cladode C let FC (Γ) be the complex of subforests of Γ contained in C.
Clearly FC (Γ) ≃ S VC −2 , where VC is the number
P of vertices of C. Since F (Γ) is the join
of all the FC (Γ), we get FC (Γ) ≃ S d−1 for d = C (VC − 1). Now, VC − 1 is the number of
non-base vertices of C, and
P every vertex of Γ except for the basepoint is a non-base vertex
of a unique cladode, so C (VC − 1) = V − 1.

Definition 4.3 (Complex of ascending forests). The complex of ascending forests F ↑ (Γ, p, ρ)
for a symmetric marked cactus graph [Γ, p, ρ] is the full subcomplex of F (Γ) supported on
those 0-simplices F such that [Γ/F, pF , ρF ] ∈ lk↑ [Γ, p, ρ].
↑
Observation 4.4. F ↑ (Γ, p, ρ) ∼
= lk [Γ, p, ρ].
d

Proof. The isomorphism is given by F 7→ [Γ/F, pF , ρF ].



For certain characters, F ↑ (Γ, p, ρ) is guaranteed to be contractible. We call these characters decisive:
Definition 4.5 (Decisive). Call a character χ of PΣAutn decisive if every [Γ, p, ρ] lies in
a unique star of a symmetric marked rose with maximal χ value.
Observation 4.6. Let χ be decisive. Then for any Γ 6= Rn , F ↑ (Γ, p, ρ) is contractible.
Proof. Every ascending forest is contained in an ascending spanning tree, and we are assuming there is a unique ascending spanning tree, so F ↑ (Γ, p, ρ) is just the star in F ↑(Γ, p, ρ)
of this unique ascending spanning tree.

Proposition 4.7 (Positive implies decisive). Positive characters of PΣAutn are decisive.
Proof. Let T be the spanning tree in Γ such that, using the notation from Lemma 3.12,
for each cladode Cj,ρ, the origin of the edge in Ej,T is the base of Cj,ρ; see Figure 2 for
an example. Note that the edge of Ej,T is before all the other edges of Cj,ρ. We claim
that ρT has larger χ value than ρU for any other spanning tree U. First we prove this in
the case when U differs from T only in one cladode, say Cj,ρ. Let Ij be the set of i such
that the projection projCj,ρ (Ci,ρ ) lies between Ej,U and Ej,T . Since Ej,T is before Ej,U , by
P
Lemma 3.12 we get hχ ([Γ/T, pT , ρT ]) − hχ ([Γ/U, pU , ρU ]) = i∈Ij ai,j > 0. Now suppose U
differs from T in more than one cladode, say Cj1 ,ρ , . . . , Cjr ,ρ . By
Ejk ,T to Ejk ,U
P
Prchanging
one k at a time, we get hχ ([Γ/T, pT , ρT ]) − hχ ([Γ/U, pU , ρU ]) = k=1 i∈Ij ai,jk > 0. We
k

conclude that hχ ([Γ/T, pT , ρT ]) > hχ ([Γ/U, pU , ρU ]), as desired.

16

MATTHEW C. B. ZAREMSKY

By a parallel argument, negative characters are also decisive.

Figure 2. A cactus graph, with the tree T from the proof of Proposition 4.7
marked in bold.
It turns out “most” characters are decisive, in the following sense:
P
Definition 4.8 (Generic). Call a character χ = i6=j ai,j χi,j of PΣAutn generic if for every
P
choice of εi,j ∈ {−1, 0, 1} we have i6=j εi,j ai,j = 0 only if εi,j = 0 for all i, j (said another
way, the ai,j have no non-trivial linear dependencies using coefficients from {−1, 0, 1}).
Observation 4.9. The set {[χ] ∈ S(PΣAutn ) | χ is generic} is dense in S(PΣAutn ).
P
Proof. Given a linear dependence i6=j εi,j ai,j = 0 with εi,j ∈ {−1, 0, 1}, the complement
of the set of character classes satisfying this dependence is open and dense in S(PΣAutn ).
Since there are only finitely many choices for the εi,j , the set of generic character classes
is also (open and) dense in S(PΣAutn ).

Proposition 4.10 (Generic implies decisive). Generic characters of PΣAutn are decisive.
Proof. Let T and U be two different spanning trees in Γ, so [Γ, p, ρ] lies in the stars of
the symmetric marked roses [Γ/T, pT , ρT ] and [Γ/U, pU , ρU ]. We claim that for χ generic,
these symmetric marked roses have different hχ values, from which the result will follow.
Using the notation from Lemma 3.12, suppose the cladodes in which T and U differ are
Cj1 ,ρ , . . . , Cjr ,ρ , so Ejk ,T 6= Ejk ,U for 1 ≤ k ≤ r, but Ej,T = Ej,U for all j 6∈ {j1 , . . . , jr }. Let
U = T0 , T1 , . . . , Tr = T be spanning trees such that we obtain Tk from Tk−1 by replacing
Ejk ,U with Ejk ,T . For each 1 ≤ k ≤ r let Ik be the set of i such that the projection
projCj ,ρ (Ci,ρ ) lies between Ejk ,T and Ejk ,U . By Lemma 3.12, we know that
k
X
hχ ([Γ/Tk , pTk , ρTk ]) − hχ ([Γ/Tk−1 , pTk−1 , ρTk−1 ]) = ±
ai,jk
i∈Ik

SYMMETRIC AUTOMORPHISMS, BNSR-INVARIANTS, AND FINITENESS PROPERTIES

17

for each 1 ≤ k ≤ r (with the plus or minus depending on whether Ejk ,TPis before or
after
P Ejk ,U ). This impliesPthat hχ ([Γ/T, pT , ρT ]) − hχ ([Γ/U, pU , ρU ]) = (± i∈I1 ai,j1 ) +
(± i∈I2 ai,j2 ) + · · · + (± i∈Ir ai,jr ). Since χ is generic, this cannot be zero.

Remark 4.11. If χ is not decisive, then F ↑ (Γ, p, ρ) is still somewhat understandable, it
just might not be contractible. A forest is ascending if and only if it lies in an ascending
spanning tree, so F ↑ (Γ, p, ρ) is the union of the stars of the ascending spanning trees. Also,
a non-empty intersection of some of these stars is again a (contractible) star, namely the
star of the forest that is the intersection of the relevant spanning trees. Hence F ↑(Γ, p, ρ)
is homotopy equivalent to the nerve of its covering by stars of ascending spanning trees.
This is isomorphic to the simplicial complex whose 0-simplices are the ascending spanning
trees, and where k of them span a (k − 1)-simplex whenever the trees have a non-empty
intersection. In theory it should be possible to compute the homotopy type of this complex,
but we are not currently interested in the non-decisive characters (since they will be totally
intractable when we study the ascending up-link in the next subsection), so we leave further
analysis of this for the future.
Since lk↑ v = lk↑d v ∗ lk↑u v and lk↑d [Γ, p, ρ] ∼
= F ↑ (Γ, p, ρ), we get:
Corollary 4.12. If χ is a decisive character of PΣAutn and v = [Γ, p, ρ] for Γ not a rose,
then lk↑ v is contractible.

4.2. Ascending up-link. Thanks to Corollary 4.12, for decisive characters of PΣAutn
the only 0-simplices of ΣKn that can have non-contractible ascending links are those of the
form [Rn , ∗, α]. These have empty down-link, so the ascending link equals the ascending
up-link. It turns out that the ascending up-link lk↑u [Rn , ∗, α] is homotopy equivalent to
a particularly nice complex In↑ (χ), the complex of ascending ideal edges, which we now
describe.
Let E(∗) be the set of half-edges of Rn incident to ∗. Since we identify π1 (Rn ) with Fn ,
the petals of Rn are naturally identified with the basis S = {x1 , . . . , xn } of Fn . We will
write i for the half-edge in the petal xi with ∗ as its origin, and i for the half-edge in xi
with ∗ as its terminus, so E(∗) = {1, 1, 2, 2, . . . , n, n}.
Definition 4.13 ((Symmetric) ideal edges). A subset A of E(∗) such that |A| ≥ 2 and
|E(∗) \ A| ≥ 1 is called an ideal edge. We say an ideal edge A splits {i, i} if {i, i} ∩ A and
{i, i} \ A are both non-empty. We call A symmetric if there exists precisely one i ∈ [n]
such that A splits {i, i}. In this case we call {i, i} the split pair of A.
Intuitively, an ideal edge A describes a way of blowing up a new edge at ∗, with the
half-edges in E(∗) \ A becoming incident to the new basepoint and the half-edges in A
becoming incident to the new non-basepoint vertex; a more rigorous discussion can be
found for example in [Jen02]. The conditions in the definition ensure that blowing up
a symmetric ideal edge results in a cactus graph. See Figure 3 for an example. The
asymmetry between the conditions |A| ≥ 2 and |E(∗) \ A| ≥ 1 arises because the basepoint

18

MATTHEW C. B. ZAREMSKY

of a cactus graph must have degree at least 2, whereas other vertices must have degree at
least 3. In fact every vertex of a cactus graph has even degree, so in practice |A| ≥ 2 is
equivalent to |A| ≥ 3 for symmetric ideal edges.

1

2

1

E(∗) \ A

1
E(∗) \ A

2

2
1
−→

A

1

2

1

2

2

1

2

1

2

−→

A

Figure 3. The symmetric ideal edge {1, 2, 2} and the non-symmetric ideal
edge {1, 2}, together with the blow-ups they produce. The former yields a
cactus graph and the latter does not.
P
Definition 4.14 (Ascending symmetric ideal edge). Let χ = i6=j ai,j χi,j be a character
of PΣAutn and let A be a symmetric ideal edge. Suppose {j, j} is the split pair of A and
let I = (A ∩ [n]) \ {j}. We call A ascending (with respect to χ) if either
P
(i) j ∈ A and i∈I ai,j > 0 or
P
(ii) j ∈ A and i∈I ai,j < 0.
For example, the symmetric ideal edge in Figure 3 is ascending if and only if a2,1 < 0. If
χ is positive (respectively negative) then A is ascending if and only if j ∈ A (respectively
j ∈ A), for {j, j}
Pthe split pair of A. If χ is generic then for any set I of pairs {i, i} and
any j ∈ [n] \ I, i∈I ai,j 6= 0, so one of A = {j} ∪ I or A = {j} ∪ I is an ascending ideal
edge.
Definition 4.15 (Compatible). Two ideal edges A and A′ are called compatible if any of
A ⊆ A′ , A′ ⊆ A or A ∩ A′ = ∅ occur.
Definition 4.16 (Complex of (ascending) symmetric ideal edges). Let In be the simplicial
complex whose 0-simplices are the symmetric ideal edges, and where a collection of symmetric ideal edges span a simplex if and only if they are pairwise compatible. Let In↑ (χ)
be the subcomplex of In spanned by the ascending symmetric ideal edges.

SYMMETRIC AUTOMORPHISMS, BNSR-INVARIANTS, AND FINITENESS PROPERTIES

19

It is a classical fact that In is homotopy equivalent to lku [Rn , ∗, α] (for any α). More
precisely, the barycentric subdivision In′ is isomorphic to lku [Rn , ∗, α]. It turns out a similar
thing happens when restricting to ascending ideal edges:
Lemma 4.17. For any character χ of PΣAutn and any 0-simplex of the form [Rn , ∗, α],
lk↑u [Rn , ∗, α] is homotopy equivalent to In↑ (χ).
Proof. Since the barycentric subdivision In′ of In is isomorphic to lku [Rn , ∗, α], we have
that lk↑u [Rn , ∗, α] is isomorphic to a subcomplex In′ (asc) of In′ . This is the subcomplex
spanned by those 0-simplices in In′ , i.e, those collections of pairwise compatible symmetric
ideal edges, whose corresponding tree blow-up makes hχ go up. Note that In↑ (χ)′ is a
subcomplex of In′ (asc), since as soon as one ideal edge in a collection corresponds to an
ascending edge blow-up the whole collection corresponds to an ascending tree blow-up.
Given a 0-simplex σ = {A1 , . . . , Ak } of In′ (asc), let φ(σ) := {Ai | Ai ∈ In↑ (χ)}. We
claim that φ(σ) is non-empty, and hence φ : In′ (asc) → In′ (asc) is a well defined map
whose image is In↑ (χ). Let [Γ, p, ρ] be the result of blowing up the ideal tree given by σ.
Let U be the spanning tree in Γ such that [Γ/U, pU , ρU ] = [Rn , ∗, α]. Since the blow-up
is ascending, the blow-down reversing it cannot be ascending, so U is not an ascending
spanning tree in Γ. Choose an ascending spanning tree T in Γ, so U 6= T . Similar to the
proof of Proposition 4.10, we can turn U into T by changing one edge at a time, and from
Lemma 3.12 we get
X
X
X
hχ ([Γ/T, pT , ρT ]) − hχ ([Γ/U, pU , ρU ]) = (±
ai,jr )
ai,j2 ) + · · · + (±
ai,j1 ) + (±
i∈I1

i∈I2

i∈Ir

where the Ik and jk are as in the proof of Proposition 4.10. SincePT is ascending but U is
not, this quantity is positive. Hence there exists k such that ± i∈Ik ai,jk > 0 (with the
“±” determined by whether Ejk ,T is before or after Ejk ,U ). Write j = jk for brevity.
Now let T ′ be the spanning tree (T \ Ej,U ) ∪ Ej,T (keep in mind that Ej,U lies in T and
not U, and Ej,T lies in U and not T ), so, roughly, T ′ is the result of changing only the part
of T in Cj,ρ to look like U. Let F := T \ Cj,ρ and consider [Γ/F, pF , ρF ]. Let E j,U and E j,T
be the images of Ej,U and Ej,T in Γ/F . The difference P
between the hχ values obtained by
blowing down E j,U versus E j,T is the positive value ± i∈Ik ai,jk from before; hence E j,U
is ascending in Γ/F and E j,T is not ascending. Now, the blow-up of [Rn , ∗, α] resulting
in [Γ/F, pF , ρF ] corresponds to one of the Ai , and E j,T is the new edge blown up. This is
an ascending blow-up, since the reverse is a non-ascending blow-down. This shows that at
least one of the Ai is indeed ascending, so φ(σ) 6= ∅.
Having shown that φ : In′ (asc) → In′ (asc) is well defined, it is easily seen to be a
poset retraction (à la [Qui78, Section 1.3]) onto its image In↑ (χ)′ , so we conclude that

lk↑u [Rn , ∗, α] ∼
= In↑ (χ).
= In′ (asc) ≃ In↑ (χ)′ ∼
It is clear from Definition 4.14 that for χ positive, the complex In↑ (χ) is independent of
χ. We will write In↑ (pos) for In↑ (χ) in this case. In Proposition 4.19 we will determine the

20

MATTHEW C. B. ZAREMSKY

connectivity properties of In↑ (pos). First we need the following useful lemma, which was
proved in [WZ16].
Lemma 4.18 (Strong Nerve Lemma). Let Y be a simplicial complex covered by subcomT
plexes Y1 , . . . , Yn . Suppose that whenever an intersection ki=1 Yji of k of them (1 ≤ k ≤ n)
is non-empty, it is (n−k −2)-connected. If the nerve N of the covering is (n−3)-connected
then so is Y . If the nerve of the covering is (n − 3)-connected but not (n − 2)-acyclic, then
so is Y .
Proof. That Y is (n − 3)-connected follows from the usual Nerve Lemma, e.g., [BLVŽ94,
Lemma 1.2], but this usual Nerve Lemma is not enough to show Y is not (n − 2)-acyclic if
the nerve is not. In [WZ16, Proposition 1.21] it was shown using spectral sequences that
indeed these hypotheses ensure that Y is not (n − 2)-acyclic.

Proposition 4.19. The complex In↑ (pos) is (n − 3)-connected but not (n − 2)-acyclic (and
hence so are lk↑u [Rn , ∗, α] and lk↑ [Rn , ∗, α] for any positive character of PΣAutn ).
Proof. We will prove that In↑ (pos) is (n − 3)-connected by using induction to prove a more
general statement, and then afterwards we will prove that In↑ (pos) is not (n − 2)-acyclic
by applying Lemma 4.18. Call a subset P ⊆ E(∗) positive if for each 1 ≤ i ≤ n we have
that i ∈ P implies i ∈ P . Define the defect d(P ) to be the number of i ∈ P with i 6∈ P .
Define the weight w(P ) of P to be the number of pairs {j, j} contained in P , plus one
if the defect is non-zero. For example the sets {1, 1, 2, 2}, {1, 2, 2} and {1, 2, 3, 4, 5, 5} all
have weight two (and defect zero, one and four, respectively). Also note that P = E(∗)
itself is positive and has defect zero and weight n. Let I ↑ (P ; pos) be the subcomplex of
In↑ (pos) supported on those 0-simplices A such that A ⊆ P . We now claim that I ↑ (P ; pos)
is (w(P ) − 3)-connected, so In↑ (pos) being (n − 3)-connected is a special case of this.
We induct on w(P ). For the base case we can use w(P ) = 1, and the result holds
vacuously since every set is (−2)-connected. Now let w(P ) ≥ 2. Let D be the set of all
i ∈ P with i 6∈ P , so d(P ) = |D|. Within this induction on w(P ) we now additionally
begin an induction on d(P ). For the base case we assume D = ∅, i.e., d(P ) = 0. Consider
the 0-simplices of I ↑ (P ; pos) of the form P \ {i}, for each i ∈ P ∩ [n]. Call these hubs, and
denote P \ {i} by Θi . Note that a symmetric ideal edge in I ↑ (P ; pos) is compatible with
a given hub if and only if it is contained in it (i.e., it cannot properly contain it nor be
disjoint from it). Any collection of pairwise compatible symmetric ideal edges in I ↑ (P ; pos)
lies in the star of some hub, so I ↑ (P ; pos) is covered by the contractible stars of the Θi .
The intersection of the stars of any k hubs, say Θi1 , . . . , Θik , is isomorphic to the complex
I ↑ (P \ {i1 , . . . , ik }; pos). For k = 1 this is contractible, being a star, and for k > 1 we
have w(P \ {i1 , . . . , ik }) = w(P ) − k + 1 < w(P ), so by induction on w(P ) we know this is
(w(P ) − k − 2)-connected. Finally, the nerve of the covering of I ↑ (P ; pos) by these stars is
the boundary of a (w(P ) − 1)-simplex, i.e., a (w(P ) − 2)-sphere, so by the first statement
in Lemma 4.18 we conclude that I ↑ (P ; pos) is (w(P ) − 3)-connected.

SYMMETRIC AUTOMORPHISMS, BNSR-INVARIANTS, AND FINITENESS PROPERTIES

21

Now suppose D 6= ∅, so d(P ) > 0. Without loss of generality we can write D =
{1, . . . , d}. We will build up to I ↑ (P ; pos) from a subcomplex with a known homotopy
type, namely the contractible star of the 0-simplex {1} ∪ (P \ D). The 0-simplices of
I ↑ (P ; pos) missing from this star are those A containing an element of {2, . . . , d} (if d = 1
there is nothing to do, so assume d ≥ 2), so to obtain I ↑ (P ; pos) from this star we will
attach these missing 0-simplices, in some order, along their relative links lkrel A. If we
can do this in an order such that the relative links are always (w(P ) − 4)-connected,
then we can conclude that I ↑ (P ; pos) is (w(P ) − 3)-connected. The order is as follows:
first glue in the A containing 2 in order of decreasing size, then the A containing 3 in
order of decreasing size, and so forth. The relative link lkrel A of A decomposes into the
out
join of its relative in-link lkin
rel A and relative out-link lkrel A. The relative in-link of A
is defined to be the subcomplex supported on those B in lkrel A such that B ⊆ A. The
relative out-link is defined to be the subcomplex supported on those B in lkrel A that
satisfy either A ⊆ B or A ∩ B = ∅. These options encompass all the ways a symmetric
ideal edge can be compatible with A, and clearly everything in lkin
rel A is compatible with
out
in
out
everything in lkrel A, so indeed lkrel A = lkrel A∗ lkrel A. To show that lkrel A is (w(P ) −4)connected for every A containing an element of {2, . . . , d}, we will consider lkin
rel A and
A
separately.
Let
{i
}
:=
A
∩
D,
so
i
∈
{2,
.
.
.
,
d},
and
let
A
:=
A
\
{i
lkout
A
A
♭
A }. The
rel
in
↑
0-simplices B in lkrel A must lie in I (A♭ ; pos), since for B to come before A in our order
while having smaller cardinality than A, it must not contain iA (so such B are actually
↑
already in the star of {1} ∪ (P \ D)). Hence lkin
rel A is isomorphic to I (A♭ ; pos), and
in
w(A♭ ) = w(A) − 1 < w(P ), so by induction on w(P ) we know lkrel A is (w(A♭ ) − 3)connected. Next, the 0-simplices B in lkout
rel A must be disjoint from {iA +1, . . . , d} and either
properly contain A or be disjoint from A. The map B 7→ B \ A♭ induces an isomorphism
↑
from lkout
rel A to I (P \(A♭ ∪{iA +1, . . . , d}); pos); the inverse map sends C to itself if iA 6∈ C
and to C ∪ A♭ if iA ∈ C. Since w(P \ (A♭ ∪ {iA + 1, . . . , d})) = w(P ) − w(A♭ ) < w(P ), by
induction on w(P ) we know lkout
rel A is (w(P ) − w(A♭ ) − 3)-connected. We conclude that
out
A
is
(((w(A
A
∗
lk
lkrel A = lkin
♭ ) − 3) + (w(P ) − w(A♭ ) − 3)) + 2)-connected, which is
rel
rel
to say (w(P ) − 4)-connected, as desired.
This finishes the inductive proof, which in particular shows In↑ (pos) is (n − 3)-connected.
Now to see that it is not (n − 2)-acyclic, consider the covering of In↑ (pos) by the stars of
Θ1 , . . . , Θn , as above. The intersection of any k of these stars is (n − k − 2)-connected, as
was deduced during the inductive proof, and the nerve of the covering is an (n − 2)-sphere,
so Lemma 4.18 says I ↑ (P ; pos) is not (n − 2)-acyclic.

A parallel argument shows that In↑ (χ) is also (n − 3)-connected but not (n − 2)-acyclic
for χ a negative character of PΣAutn .
Remark 4.20. If χ is neither positive nor negative then In↑ (χ) is much more complicated,
for example as discussed in Remark 4.24 below, one can find examples of generic χ for
which I4↑ (χ) has non-trivial π1 and H2 . Hence, we have focused only on positive and
negative characters of PΣAutn in Theorem A, but in Subsection 4.3 we will show that
generic characters are also tractable at least when n = 3.

22

MATTHEW C. B. ZAREMSKY

We can now prove Theorem A.
Proof of Theorem A. By Corollary 4.12 and Proposition 4.19, all the ascending links of
0-simplices in ΣKn are (n − 3)-connected, so Corollary 1.7 says the filtration (ΣKnχ≥t )t∈R
is essentially (n − 3)-connected, and so [χ] ∈ Σn−2 (PΣAutn ).
To prove the negative statement, note that Proposition 4.19 says that there exist ascending links of 0-simplices that are not (n − 2)-acyclic, with arbitrary hχ value. Also,
every ascending link has trivial (n − 1)st homology since it is (n − 2)-dimensional. By
Corollary 1.7 then, the filtration (ΣKnχ≥t )t∈R is not essentially (n − 2)-connected, and so
[χ] 6∈ Σn−1 (PΣAutn ).

Remark 4.21. Using the natural
P split epimorphisms PΣAutn → PΣAutm for m < n,
we also now can see that if χ = i6=j ai,j χi,j is a character of PΣAutn induced from this
epimorphism by a positive or negative character of PΣAutm (so ai,j is positive for 1 ≤
i, j ≤ m or negative for all 1 ≤ i, j ≤ m, and is zero if either i or j is greater than m) then
[χ] ∈ Σm−2 (PΣAutn ). However, we cannot immediately tell whether [χ] 6∈ Σm−1 (PΣAutn )
(which we suspect is the case), since Pettet showed the kernels of PΣAutn → PΣAutm
have bad finiteness properties [Pet10].
As an immediate consequence of Theorem A, Citation 1.2 and Observation 2.3, we have
the following result, which will provide the crucial step in proving Theorem B in Section 5.
Corollary 4.22. For n ≥ 2, if χ is a discrete positive character of PΣAutn , then the kernel
ker(χ) is of type Fn−2 but not Fn−1 . In particular the “Bestvina–Brady-esque” subgroup
BBn , i.e., the kernel of the character sending each standard generator αi,j to 1 ∈ Z, is of
type Fn−2 but not Fn−1 .

4.3. The n = 3 case. When n = 3 the 0-simplex links in ΣK3 are graphs, and using some
graph theoretic considerations we can actually prove the analog of Proposition 4.19 for
generic characters, which leads to the following:
Theorem 4.23. Σ2 (PΣAut3 ) = ∅.
Proof. Since Σ2 (PΣAut3 ) is open and the generic character classes are dense in the character sphere (Observation 4.9), it suffices to prove the analog of Proposition 4.19 for all
generic χ. Since I3↑ (χ) is 1-dimensional, i.e., a graph, we need to prove that it is connected
but not a tree.
First we collect some facts about I3 . It is a graph with eighteen vertices, namely there
are twelve vertices corresponding to the symmetric ideal edges A ⊆ E(∗) = {1, 1, 2, 2, 3, 3}
with |A| = 3 (three choices of which {i, i} to split, times two choices of which of i or i to
include in A, times two choices of which {j, j} (j 6= i) to include in A) and six vertices for
the symmetric ideal edges with |A| = 5 (six choices of which element of E(∗) to leave out
of A). Call the former vertices depots and the latter hubs. There is an edge connecting a
depot to a hub whenever the depot is contained in the hub, and there is an edge connecting

SYMMETRIC AUTOMORPHISMS, BNSR-INVARIANTS, AND FINITENESS PROPERTIES

23

two depots whenever they are disjoint. Each depot has degree four and each hub has degree
six.
Now, since χ is generic, for each pair of depots of the form {i, j, j} and {i, j, j}, precisely
one of them is ascending, with a similar statement for hubs. Hence I3↑ (χ) is a full subgraph
of I3 spanned by six depots and three hubs. We claim that each ascending hub has degree
at least three in I3↑ (χ). Consider the hub {i′ , j, j, k, k}, where i′ ∈ {i, i} and i, j, k are
distinct. If this is ascending then at least one of the depots {i′ , j, j} or {i′ , k, k} must be
as well, since if aj,i + ak,i is positive (respectively negative) then at least one of aj,i or ak,i
must be as well. The other two edges incident to our hub come from the fact that one of
the depots {j, k, k} or {j, k, k} is ascending, as is one of {j, j, k} or {j, j, k}.
Having shown that each hub in I3↑ (χ) has degree at least three in I3↑ (χ), this tells us
I3↑ (χ) has at least nine edges, since hubs cannot be adjacent. Then since I3↑ (χ) also has
nine vertices we conclude that it contains a non-trivial cycle. It remains to prove it is
connected. Say the three hubs in I3↑ (χ) are u, v and w, and suppose that u has no adjacent
depots in I3↑ (χ) in common with either v or w (since otherwise we are done). Since there
are only six depots, this implies that v and w have the same set of adjacent depots in
I3↑ (χ). But this is impossible since the intersection of the stars of two different ascending
hubs can contain at most two ascending depots.

Combining this with Orlandi-Korner’s computation of Σ1 (PΣAut3 ), we get in particular
that Σ1 (PΣAut3 ) is dense in S(PΣAut3 ) = S 5 but Σ2 (PΣAut3 ) is already empty. (Note
since PΣAut2 ∼
= F2 we also know that Σ1 (PΣAut2 ) = ∅.)
Remark 4.24. Unfortunately the analogous result for arbitrary n, i.e., that Σn−2 (PΣAutn )
is dense in S(PΣAutn ) but Σn−1 (PΣAutn ) is empty, cannot be deduced using our methods when n > 3 (though we do suspect it is true). One would hope that the analog of
Proposition 4.19 always holds for all generic χ, but it does not. For example, when n = 4
one can find a generic character χ such that I4↑ (χ) is not simply connected. One example
we found is to take χ with a1,2 = a2,1 = a3,4 = a4,3 = 3 and all other ai,j = −1 (adjusted
slightly by some tiny ε > 0 to be generic). This non-simply connected ascending link
also has non-trivial second homology, so this does not necessarily mean that [χ] is not
in Σ2 (PΣAut4 ) (and we believe that it actually is), it is just inconclusive. In general we
tentatively conjecture that Σn−2 (PΣAutn ) is dense in S(PΣAutn ) and Σn−1 (PΣAutn ) is
empty, but for now our Morse theoretic approach here seems to only be able to handle the
positive and negative characters for arbitrary n, and also the generic characters for n = 3.
5. Proof of Theorem B
We can now use our results about PΣAutn to quickly prove Theorem B, about ΣAutn .
Proof of Theorem B. Since BBn is of type Fn−2 but not Fn−1 by Corollary 4.22, and has
finite index in ΣAut′n by Lemma 2.5, we know that ΣAut′n is of type Fn−2 but not Fn−1 .

24

MATTHEW C. B. ZAREMSKY

Also, Observation 2.3 says that for any m either Σm (ΣAutn ) is all of S 0 or else is empty.
The result now follows from Citation 1.2.

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Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902
E-mail address: zaremsky@math.binghamton.edu

