COUNTING CONJUGACY CLASSES IN Out(FN )

arXiv:1707.07095v1 [] 22 Jul 2017

MICHAEL HULL AND ILYA KAPOVICH

Abstract. We show that if a f.g. group G has a non-elementary WPD action on a hyperbolic
metric space X, then the number of G-conjugacy classes of X-loxodromic elements of G coming
from a ball of radius R in the Cayley graph of G grows exponentially in R. As an application we
prove that for N ≥ 3 the number of distinct Out(FN )-conjugacy classes of fully irreducibles ϕ from
an R-ball in the Cayley graph of Out(FN ) with log λ(ϕ) on the order of R grows exponentially in
R.

1. Introduction
The study of the growth of the number of periodic orbits of a dynamical system is an important
theme in dynamics and geometry. A classic and still incredibly influential result of Margulis [19]
computes the precise asymptotics of the number of closed geodesics of length ≤ R (that is, of
periodic orbits of geodesic flow of length ≤ R) for a compact hyperbolic manifold. A recent famous
result of Eskin and Mirzakhani [11], which served as a motivation for this note, shows that for
the moduli space Mg of a closed oriented surface Sg of genus g ≥ 2, the number N (R) of closed
hR
Teichmuller geodesics in Mg of length ≤ R grows as N (R) ∼ ehR as R → ∞ where h = 6g −6. Note
that in the context of a group action, counting closed geodesics amounts to counting conjugacy
classes of elements rather than counting elements displacing a basepoint by distance ≤ R. The
problems about counting conjugacy classes with various metric restrictions are harder than those
about counting group elements and many group-theoretic tricks (e.g. embeddings of free subgroups
or of free subsemigroups) do not, a priori, give any useful information about the growth of the
number of conjugacy classes. In the context of the Eskin-Mirzakhani result mentioned above, a
closed Teichmuller geodesic of length ≤ R in Mg comes from an axis L(ϕ) in the Teichmüller
space Tg of a pseudo-Anosov element ϕ ∈ M od(Sg ) with translation length τ (ϕ) ≤ R. Note that
τ (ϕ) = log λ(ϕ), where λ(ϕ) is the dilatation of ϕ. Thus N (R) is the number of M od(Sg )-conjugacy
classes of pseudo-Anosov elements ϕ ∈ M od(Sg ) with log λ(ϕ) ≤ R, where M od(Sg ) is the mapping
class group.
In this note we study a version of this question for Out(FN ) where N ≥ 3. The main analog of
being pseudo-Anosov in the Out(FN ) context is the notion of a fully irreducible element. Every fully
irreducible element ϕ ∈ Out(FN ) has a well-defined stretch factor λ(ϕ) > 1 (see [5]) which is an
invariant of the Out(FN )-conjugacy class of ϕ. For specific ϕ one can compute λ(ϕ) via train track
methods, but counting the number of distinct λ(ϕ) with log(λ(ϕ)) ≤ R (where ϕ ∈ Out(FN ) is
fully irreducible) appears to be an unapproachable task. Other Out(FN )-conjugacy invariants such
as the index, the index list and the ideal Whitehead graph of fully irreducibles [6, 7, 14, 21], also do
not appear to be well suited for counting problems. Unlike the Teichmüller space, the Outer space
2010 Mathematics Subject Classification. Primary 20F65, Secondary 57M, 37B, 37D.
The second author was supported by the individual NSF grants DMS-1405146 and DMS-1710868. Both authors
acknowledge the support of the conference grant DMS-1719710 “Conference on Groups and Computation”.
1

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MICHAEL HULL AND ILYA KAPOVICH

CVN does not have a nice local analytic/geometric structure similar to the Teichmuller geodesic
flow. Moreover, as we explain in Remark 3.2 below, it is reasonable to expect that (for N ≥ 3)
the number of Out(FN )-conjugacy classes of fully irreducibles ϕ ∈ Out(FN ) with log λ(ϕ) probably
grows as a double exponential in R (rather than as a single exponential in R, as in the case of Mg ).
Therefore, to get an exponential growth of the number of conjugacy classes one needs to consider
more restricted context, namely where the elements come from an R-ball in the Cayley graph of
Out(FN ).
Here we obtain:
Theorem 1.1. Let N ≥ 3. Let S be a finite generating set for Out(FN ). There exist constants
σ > 1 and C2 > C1 > 0, R0 ≥ 1 such that the following hols. For R ≥ 0 let BR be the ball of radius
R in the Cayley graph of Out(FN ) with respect to S.
Denote by cR the number of distinct Out(FN )-conjugacy classes of fully irreducible elements
ϕ ∈ BR such that C1 R ≤ log λ(ϕ) ≤ C2 R. Then
cR ≥ σ R
for all R ≥ R0 .
In the process of the proof of Theorem 1.1 we establish the following general result:
Theorem 1.2. Suppose G has a cobounded WPD action on a hyperbolic metric space X and L is
a non-elementary subgroup of G for this action. Let S be a generating set of G. For R ≥ 1 let bR
be the number of distinct G-conjugacy classes of elements of g ∈ L that act loxodromically on X
and such that |g|S ≤ R. Then there exist R0 ≥ 1 and c > 1 such that for all R ≥ R0
bR ≥ cR .
This statement is mostly relevant in the case where G is finitely generated and S is finite, but the
conclusion of Theorem 1.2 is not obvious even if S is infinite. Theorem 1.2 is a generalization of [16,
Theorem 1.1], which states (under a different but equivalent hypothesis, see [20]) that such G has
exponential conjugacy growth. The proofs of Theorem 1.2 and [16, Theorem 1.1] are similar and are
both based on the theory of hyperbolically embedded subgroups developed in [9], and specifically
the construction of virtually free hyperbolically embedded subgroups in [Theorem 6.14][9].
Both Theorem 1.1 and Theorem 1.2 are derived using:
Theorem 1.3. Suppose G has a cobounded WPD action on a hyperbolic metric space X and L
is a non-elementary subgroup of G for this action. Then for every n ≥ 2 there exists a subgroup
H ≤ L and a finite group K ≤ G such that:
(1) H ∼
= Fn .
(2) H × K ֒→h G.
(3) The orbit map H → X is a quasi-isometric embedding.
In particular, every element of H is loxodromic with respect to the action on X and two elements
of H are conjugate in G if and only if they are conjugate in H.
Here for a subgroup A of a group G writing A ֒→h G means that A is hyperbolically embedded
in G.
The proof of Theorem 1.1 also uses, as an essential ingredient, a result of Dowdall and Taylor
[10, Theorem 4.1] about quasigeodesics in the Outer spaces and the free factor complex. Note

COUNTING CONJUGACY CLASSES IN Out(FN )

3

that it is still not known whether the action of Out(FN ) on the free factor complex is acylindrical,
and, in a sense, the use of Theorem 1.2 provides a partial workaround here. Note that in the
proof of Theorem 1.1 instead of using Theorem 1.3 we could have used an argument about stable
subgroups having finite width. It is proved in [2] that convex cocompact (that is, finitely generated
and with the orbit map to the free factor complex being a quasi-isometric embedding) subgroups
of Out(FN ) are stable in Out(FN ). In turn, it is proved in [1] that if H is a stable subgroup of a
group G, then H has finite width in G (see [1] for the relevant definitions). Having finite width is a
property sufficiently close to being malnormal to allow counting arguments with conjugacy classes
to go through. The proof given here, based on using Theorem 1.3 above, is different in flavor and
proceeds from rather general assumptions. Note that in the conclusion of Theorem 1.3 the fact
that H × K ֒→h G implies that H and H × K are stable in G.
Another possible approach to counting conjugacy classes involves the notion of “statistically
convex cocompact actions” recently introduced and studied by Yang, see [22, 23] (particularly
[23, Theorem B] about genericity of conjugacy classes of strongly contracting elements). However,
Yang’s results only apply to actions on proper geodesic metric spaces with finite critical exponent
for the action, such as, for example, the action of the mapping class group on the Teichmüller space.
For essentially the same reasons as explained in Remark 3.2 below, the action of Out(FN ) (where
N ≥ 3) on CVN , with either the asymmetric or symmetrized Lipschitz metric, has infinite critical
exponent. Still, it is possible that the statistical convex cocompacness methods may be applicable
to the actions on CVN of some interesting subgroups of Out(FN ).
The first author would like to thank the organizers of the conference “Groups and Computation”
at Stevens Institute of Technology, June 2017. The second author is grateful to Spencer Dowdall,
Samuel Taylor, Wenyuan Yang and Paul Schupp for helpful conversations.
2. Conjugacy classes of loxodromics for WPD actions
We assume throughout that all metric spaces under consideration are geodesic and all group
actions on metric spaces are by isometries. Given a generating set A of a group G, we let Cay(G, A)
denote the Cayley graph of G with respect to A. In order to directly apply results from [9] it is
more convenient to consider actions on Cayley graphs. By the well known Milnor-Svarc Lemma
this is equivalent to considering cobounded actions.
Lemma 2.1 (Milnor-Svarc). If G acts coboundedly on a geodesic metric space X, then there exists
A ⊆ G such that Cay(G, A) is G-equivarently quasi-isometric to X.
For a subgroup H ≤ G and a subset S ⊆ G, we use H ֒→h (G, S) to denote that H is hyperbolically
embedded in G with respect to S, or simply H ֒→h G if H ֒→h (G, S) for some S ⊆ G. We refer to
[9] for the definition of a hyperbolically embedded subgroup. The only property of a hyperbolically
embedded subgroup H that we use is that H is almost malnormal, that is for g ∈ G \ H, the
intersection of H and H g is finite [9, Proposition 4.33]. Note that this implies that any two infinite
order elements of H are conjugate in G if and only if they are conjugate in H.
For a metric space X and an isometry g of X, the asymptotic translation length ||g||X is defined
i
as ||g||X = limi→∞ d(g ix,x) , where x ∈ X. It is well-known that this limit always exists and is
independent of x ∈ X. If ||g||X > 0 then g is called loxodromic. For a group G acting on X, a
loxodromic element is called a WPD element if for all ε > 0 and all x ∈ X, there exists N ∈ N
such that
|{h ∈ G | d(x, hx) < ε, d(g n x, hg n x) < ε}| < ∞.

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MICHAEL HULL AND ILYA KAPOVICH

We say the action of G on X is WPD if every loxodromic element is a WPD element.
Now we fix a subset A ⊆ G such that Cay(G, A) is hyperbolic and the action of G on Cay(G, A) is
WPD. We say g ∈ G is loxodromic if it is loxodromic with respect to the action of G on Cay(G, A).
Each such element is contained in a unique, maximal virtually cyclic subgroup E(g) [9, Lemma
6.5].
Lemma 2.2. [15, Corollary 3.17] If g1 , ..., gn are non-commensurable loxodromic elements, then
{E(g1 ), ..., E(gn )} ֒→h (G, A).
A subgroup L ≤ G is called non-elementary if L contains two non-commensurable loxodromic
elements. Let KG (L) denote the maximal finite subgroup of G normalized by L. When L is
non-elementary, this subgroup is well-defined by [15, Lemma 5.5].
The following lemma was proved in [15] under the assumption that the action is acylindrical,
but the proof only uses that the action is WPD.
Lemma 2.3. [15, Lemma 5.6] Let L be a non-elementary subgroup of G. Then there exists noncommensurable, loxodromic elements g1 , ..., gn contained in L such that E(gi ) = hgi i × KG (L).
Proof of Theorem 1.3. First we note that (3) implies that every element of H is loxodromic with
respect to the action on X. Also, the fact that two elements of H are conjugate in G if and only
if they are conjugate in H follows from the fact that H × K is almost malnormal in G and K acts
trivially on H by conjugation.
We use the construction from [Theorem 6.14][9]. As in [Theorem 6.14][9], we let n = 2 since the
construction from this case can be easily modified for any n.
By Lemma 2.1, we can assume X = Cay(G, A) for some A ⊆ G. Let g1 , ..., g6 be elements of
L given by Lemma 2.3. Then each E(gi ) = hgi i × KG (L) and furthermore {E(g1 ), ..., E(g6 )} ֒→h
(G, A) by Lemma 2.2. Let
6
[
E(gi ) \ {1}.
E=
g1n g2n g3n

i=1
g4n g5n g6n

for sufficiently large n. It is shown in [9] that x
and y =
Let H = hx, yi where x =
and y generate a free subgroup of G and this subgroup is quasi-convex in Cay(G, A ∪ E). Hence H
(with the natural word metric) is quasi-isometrically embedded in Cay(G, A∪E), and since the map
Cay(G, A) → Cay(G, A ∪ E) is 1-lipschitz H is also quasi-isometrically embedded in Cay(G, A).
Let K = KG (L). Since x and y both commute with K, hH, Ki ∼
= H × K. Finally, we can apply [9,
Theorem 4.42] to get that H × K ֒→h G. Verifing the assumptions of [9, Theorem 4.42] is identical
to the proof of [9, Theorem 6.14].

Note that Theorem 1.2 is an immediate consequence of Theorem 1.3.
3. The case of Out(FN ).
We assume familiarity of the reader with the basics related to Out(FN ) and Outer space. For
the background information on these topics we refer the reader to [4, 5, 8, 12].
In what follows we assume that N ≥ 2 is an integer, CVN is the (volume-one normalized) CullerVogtmann Outer space, FN is the free factor complex for FN , dC is the asymmetric Lipschitz metric
on CVN and dsym
is the symmetrized Lipschitz metric on CVN . When we talk about the Hausdorff
C

COUNTING CONJUGACY CLASSES IN Out(FN )

5

distance dHaus in CVN , we always mean the Hausdorff distance with respect to dsym
C . For K ≥ 1,
by a K-quasigeodesic in CVN we mean a function γ : I → CVN (where I ⊆ R) is an interval, such
that for all s, t ∈ I, s ≤ t we have
1
(t − s) − K ≤ dC (γ(s), γ(t)) ≤ K(t − s) + K.
K
For ǫ > 0 we denote by CVN,ǫ the ǫ-thick part of CVN .
We recall a portion of one of the main technical results of Dowdall and Taylor, [10, Theorem
4.1]:
Proposition 3.1. Let K ≥ 1 and let γ : R → CVN be a K-quasigeodesic such that its projection
π ◦ γ : R → FN is also a K-quasigeodesic. There exist constants D > 0, ǫ > 0, depending only on
K and N , such that the following holds:
If ρ : R → CVN is any geodesic with the same endpoints as γ then:
(1) We have γ(R), ρ(R) ⊂ CVN,ǫ .
(2) We have dHaus (γ(R), ρ(R)) ≤ D.
Here saying that γ and ρ have the same endpoints means that supt∈R dsym
C (γ(t), ρ(t)) < ∞.
Proof of Theorem 1.1. By [4], the action of Out(FN ) on FN satisfies the hypothesis of Theorem 1.3.
Let H be the subgroup provided by Theorem 1.3 with L = Out(Fn ).
We fix a free basis A = {a, b} for the free group H, and let dA be the corresponding word metric
on H.
Note that the assumptions on H imply that every nontrivial element of H is fully irreducible.
Moreover, if we pick a base-point p in FN , then there is K ≥ 1 such that the image of every geodesic
in the Cayley graph Cay(H, A) in FN under the orbit map is a (parameterized) K-quasigeodesic.
Pick a basepoint G0 ∈ CVN . Since the projection π : (CVN , dC ) → FN is coarsely Lipschitz [4],
and since the orbit map H → FN is a quasi-isometric embedding, it follows that the orbit map
(H, dA ) → (CVN , dC ), u 7→ uG0 is a K1 -quasi-isometric embedding for some K1 ≥ 1. Moreover,
the image of this orbit map lives in an ǫ0 -thick part of CVN (where ǫ0 is the injectivity radius of
G0 ). Since on CVN,ǫ0 the metrics dC and dsym
are bi-Lipschitz equivalent, it follows that the orbit
C
sym
map (H, dA ) → (CVN , dC ), u 7→ uG0 is a K2 -quasi-isometric embedding for some K2 ≥ 1. For
every c ∈ A±1 fix a dC -geodesic τc from G0 to cG0 .
Now let γ : I → Cay(H, A) be a geodesic such that γ −1 (H) = I ∩ Z and such that the endpoints
of I (if any) are integers. We then define a path γ : I → CVN as follows. Whenever n ∈ Z is such
that [n, n + 1] ⊆ I, then γ(n) = g and γ(n + 1) = gc for some c ∈ A±1 . In this case we define
γ|[n,n+1] to be gτc . Then for every geodesic γ : I → Cay(H, A) as above the path γ : I → CVN is
K3 -quasigeodesic, with respect to both dC and dsym
C , for some K3 ≥ 1 independent of γ. Moreover,
γ(I) ⊂ CVN,ǫ1 for some ǫ1 > 0 independent of γ.
Let w be a cyclically reduced word of length n ≥ 1 in H. Consider the bi-infinite w−1 -periodic
geodesic γ : R → Cay(H, A) with γ(0) = 1 and γ(n) = w−1 . Thus the path the path γ : I → CVN
is K3 -quasigeodesic, with respect to both dC and dsym
C , and γ(I) ⊂ CVN,ǫ1 . Since 1 6= w ∈ H,
it follows that w is fully irreducible as an element of Out(FN ). Hence w can be represented by
an expanding irreducible train-track map f : G → G with the Perron-Frobenius eigenvalue λ(f )
equal to the stretch factor λ(w) of the outer automorphism w ∈ Out(FN ). There exists a volumeone “eigenmetric” df on G with respect to which f is a local λ(f )-homothety. Then, if we view

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MICHAEL HULL AND ILYA KAPOVICH

(G, df ) as a point of CVN , then dC (G, Gw) = log λ(w). Moreover, in this case we can construct
a w-periodic dC -geodesic folding line ρ : R → CVN with the property that for any integer i,
ρ(i) = Gwi , hence for integers i < j dC (Gwi , Gwj ) = (j − i) log λ(w). Thus, for any i > 0 we have
dC (G, w−i G) = i log λ(w).
The bi-infinite lines ρ and γ are both w−1 -periodic (in the sense of the left action of w−1 ) and
therefore supt∈R dsym
C (γ(t), ρ(t)) < ∞.
Hence, by Proposition 3.1, there exist constants D > 0 and ǫ > 0 (independent of w) such
that ρ ⊂ CVN,ǫ and that dHaus (ρ, γ)) ≤ D. The fact that ρ ⊂ CVN,ǫ implies that ρ is a K4 quasigeodesic with respect to dsym
for some constant K4 ≥ 1 independent of w. Now for the
C
asymptotic translation length ||w||CV , where w is viewed as an isometry of (CVN , dsym
C ), we get
that one one hand (using the line ρ)
1
log λ(w−1 ) ≤ ||w||CV ≤ K4 log λ(w−1 )
K4
and on the other hand (using the line γ) that
1
n ≤ ||w||CV ≤ K3 n.
K3
Therefore
1
n ≤ log λ(w−1 ) ≤ K3 K4 n.
K3 K4
Recall also, that by a result of Handel and Mosher [13], there exists a constant M = M (N ) ≥ 1
log λ(ϕ)
1
≤ log
such that for every fully irreducible ϕ ∈ Out(FN ) we have M
λ(ϕ−1 ) ≤ M . Therefore for w as
above we have
1
n ≤ log λ(w) ≤ K3 K4 M n.
K3 K4 M
Recall that A = {a, b} and that S is a finite generating set for Out(FN ). Put M ′ =
max{|a|S , |b|S }, so that for every freely reduced word w in H we have |w|S ≤ M ′ |w|A .
R
For R >> 1 put n = ⌊ M
′ ⌋. The number of distinct H-conjugacy classes represented by cyclically
reduced words w of length n is ≥ 2n for n big enough. Recall two elements of H are conjugate in
H of and only if they are conjugate in Out(FN ). Therefore we get
R

≥ 2n ≥ 2 M ′ −1
distinct Out(FN )-conjugacy classes from such words w. As we have seen above, each such w gives
us a fully irreducible element of Out(FN ) with
R
R
1
≤ log λ(w) ≤ K3 K4 M n ≤ K3 K4 M ′ ,
′
K3 K4 M 2M
M
and the statement of Theorem 1.1 is verified.



Remark 3.2. As noted in the introduction, unlike in the mapping class group case, we expect
that for N ≥ 3 the number of Out(FN )-conjugacy classes of fully irreducibles ϕ ∈ Out(FN ) with
log λ(ϕ) ≤ R grows as a double exponential in R. A double exponential upper bound follows from

COUNTING CONJUGACY CLASSES IN Out(FN )

7

general Perron-Frobenius considerations. Every fully irreducible ϕ ∈ Out(FN ) can be represented
by an expanding irreducible train track map f : G → G, where G is a finite connected graph with
b1 (G) = N and with all vertices in G of degree ≤ 3. The number of possibilities for such G is finite
in terms of N . By [18, Proposition A.4], if f is as above and λ := λ(f ), then max mij ≤ kλk+1
(where k is the number of edges in G and M = (mij )ij is the transition matrix of f ). If log λ ≤ R,
we get max log mij ≤ log k + (k + 1)R and max mij ≤ ke(k+1)R . Thus we get exponentially many
(in terms of R) possibilities for the transition matrix M of f . Since for a given length L there
are exponentially many paths of length L in G, this yields an (a priori) double exponential upper
bound for the number of train track maps representing fully irreducible elements of Out(FN ) with
log λ ≤ R.
For the prospective double exponential lower bound we give the following explicit construction
for the case of F3 . Let w ∈ F (b, c) be a nontrivial positive word containing the subwords b2 , c2 ,
bc and cb. Consider the automorphism ϕw of F3 = F (a, b, c) defined as ϕw (a) = b, ϕw (b) = c,
ϕw (c) = aw(b, c). We can also view ϕw as a graph map fw : R3 → R3 where R3 is the 3-rose with
the petals marked by a, b, c. Then fw is an expanding irreducible train track map representing
ϕw . Moreover, a direct check shows that, under the assumptions made on w, the Whitehead graph
of fw is connected. Additionally, for a given n ≥ 1, “most” positive words of length n in F (b, c)
satisfy the above conditions and define fully irreducible automorphisms ϕw . To see this, observe
that the free-by-cyclic group Gw = F3 ⋊ϕw Z can be rewritten as a one-relator group:
Gw = ha, b, c, t|t−1 at = b, t−1 bt = c, t−1 ct = aw(b, c)i =
ha, t|t−3 at3 = aw(t−1 at, t−2 at2 )i.
Moreover, one can check that if w was a C ′ (1/20) word, then the above one-relator presentation
of Gw satisfies the C ′ (1/6) small cancellation condition, and therefore Gw is word-hyperbolic and
the automorphism ϕw ∈ Out(F3 ) is atoroidal. Since, as noted above, ϕw admits an expanding
irreducible train track map on the rose with connected Whitehead graph, a result of Kapovich [17]
implies that ϕ is fully irreducible. Moreover, if |w| = L, then it is not hard to check that log λ(fw ) =
log λ(ϕw ) grows like log L.
Since “random” positive words w ∈ F (b, c) are C ′ (1/20) and contain b2 , c2 , cb, bc as subwords, for
sufficiently large R ≥ 1, the above construction produces doubly exponentially many atoroidal fully
irreducible automorphisms ϕw with |w| = eR and log λ(ϕw ) on the order of R. We conjecture that
in fact most of these elements are pairwise non-conjugate in Out(F3 ) and that this method yields
doubly exponentially many fully irreducible elements ϕ of Out(F3 ) with log λ(ϕ) ≤ R. However,
verifying this conjecture appears to require some new techniques and ideas beyond the reach of this
paper.
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Department of Mathematics, PO Box 118105, University of Florida, Gainesville, FL 32611-8105
https://people.clas.ufl.edu/mbhull/
E-mail address: mbhull@ufl.edu
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street,
Urbana, IL 61801, USA
http://www.math.uiuc.edu/~kapovich/
E-mail address: kapovich@math.uiuc.edu

