JSJ VI: Growth of automorphisms

This post is based on a talk given by Stefanie Zbinden.

In this talk we will see how JSJ theory relates to length and growth of elements in a group $G$, which is a subject that has received a lot of attention and remains an active area of research even in the case of free and surface groups. Our notion of length is the conjugacy length $\|g\|$, i.e. the minimum word length of an element in the conjugacy class of $g$, fixing once and for all a finite generating set of $G$, whose choice will play no role. If $G$ is the fundamental group of a compact Riemannian manifold, then $\|g\|$ is roughly equal to the length of a shortest closed geodesic in the free homotopy class determined by $g$. We then define the growth rate of $g$ under $\phi$ as the equivalence class of the sequence $n\mapsto\|\phi^n(g)\|$ up to bi-Lipschitz equivalence. More crudely, one can define the stretch factor of $\phi$ as

\[ {\mathrm str}(\phi):=\sup_{g\in G}\,\limsup_{n\rightarrow+\infty}\,\|\phi^n(g)\|^{1/n} . \]

The first and most important groups for which automorphism growth came to be fully understood were free abelian groups $\mathbb{Z}^m$, surface groups $\pi_1(S)$, and free groups $F_m$. Growth of elements of $\mathrm{Out}(\mathbb{Z}^m)={\mathrm GL}_m(\mathbb{Z})$ can be easily described in terms of the Jordan decomposition, while growth of elements of the mapping class group of a compact surface $S$ is completely encoded in the corresponding Nielsen--Thurston decomposition. Finally, analysing growth of general automorphism of free groups $F_m$ proved to be a significantly more complex problem, which required the development of refined techniques inspired by train tracks on surfaces and was finally solved by Levitt. In all of these examples, each outer automorphism admits only finitely many growth rates as the element $g$ varies in $G$, and each growth rate is bi-Lipschitz equivalent to a sequence $n\mapsto n^p\lambda^n$ for some $p\in\mathbb{N}$ and $\lambda\geq 1$. Moreover, the number $\lambda$ happens to be an algebraic integer and a weak Perron number (i.e. it has maximal modulus among its Galois conjugates).

Automorphisms of Gromov-hyperbolic groups (and toral relatively hyperbolic groups) have a similar growth behaviour. At the same time, almost nothing is known on growth of automorphisms of non-hyperbolic groups and all classical techniques to approach this problem --- mainly train tracks and JSJ decompositions --- are known to fail or run into serious issues when one abandons the world of (relatively) hyperbolic groups. Even restricting to the rather tame world of right-angled Artin groups, any growth information seems to be available only under significant restrictions on the type of automorphism under consideration. 

What is more, Coulon recently constructed a menagerie of finitely generated groups whose automorphisms exhibit a variety of ``intermediate growth'' behaviours: they growth faster than any polynomial and slower than any exponential. It remains unknown if these exotic behaviours can occur for automorphisms of finitely presented groups, but there seems to be no reason to expect the contrary.

One of the many important results proved in Elia's paper is that growth of automorphisms of special groups is rather well-behaved, at least when it comes to the top growth rate. As a particular case, this applies to all automorphisms of right-angled Artin groups.

Theorem 1: Let $G$ be a virtually special group. The following hold for every $\phi\in\mathrm{Out}(G)$. 

•  The stretch factor ${\mathrm str}(\phi)$ is an algebraic integer and a weak Perron number.
•  If ${\mathrm str}(\phi)=1$, then $\phi$ grows at most polynomially.

This is a difficult theorem which we won't have time to cover in full, but we can see the starting step, which requires finding a clever way to apply the results of the previous talk.

If $a,b\colon\mathbb{N}\rightarrow\mathbb{R}_{>0}$ are two sequences, we write $a\preceq b$ if there exists a constant $C$ such that $a_n\leq Cb_n$ for all $n\geq 0$; equivalently, if we have $\limsup_n a_n/b_n<+\infty$. Informally, we will also say that $a$ is slower than $b$, and that $b$ is faster than $a$. We say that $a$ and $b$ are equivalent, written $a\sim b$, if we have both $a\preceq b$ and $b\preceq a$. We denote by $[1]$ the equivalence class of constant sequences.

Definition:   A growth rate is a $\sim$--equivalence class $[a_n]$ of sequences in $(\mathbb{R}_{>0})^{\mathbb{N}}$ with $[a_n]\succeq [1]$. We denote by $(\mathfrak{G},\preceq)$ the set of all growth rates with the poset structure induced by the relation $\preceq$.

Definition:  Consider some $\phi\in\mathrm{Out}(G)$ and some abstract growth rate $\mathfrak{o}\in\mathfrak{G}$. An element $g\in G$ is $\mathfrak{o}$--controlled if we have $\|\phi^n(g)\|\preceq\mathfrak{o}$. A subgroup $H\leq G$ is $\mathfrak{o}$--controlled if all its elements are $\mathfrak{o}$--controlled. We denote by $\mathcal{K}(\mathfrak{o},\phi)$ or $\mathcal{K}(\mathfrak{o})$ the family of $\mathfrak{o}$--controlled subgroups of $G$.

Consider now a finitely generated group $G$, an automorphism $\varphi\in\mathrm{Aut}(G)$ and its outer class $\phi\in\mathrm{Out}(G)$. Let $|\cdot|$ and $\|\cdot\|$ be the word and conjugacy lengths on $G$ with respect to some finite generating set, whose choice will play no role.

Definition:  The growth rate of an element $g\in G$ under $\varphi$ is the $\sim$--equivalence class $\big[|\varphi^n(g)|\big]$ in $\mathfrak{G}$. Similarly, the growth rate of $g$ under $\phi$ is\footnote{Note that $\phi^n(g)$ is not a well-defined element of $G$, but it is a well-defined conjugacy class.} the equivalence class $\big[\|\phi^n(g)\|\big]\in\mathfrak{G}$. 

In other words, we consider the equivalence class of the sequences $n\mapsto |\varphi^n(g)|$ and $n\mapsto\|\phi^n(g)\|$ up to multiplicative constants. A different choice of generating set on $G$ only alters $|\cdot|$ and $\|\cdot\|$ through a bi-Lipschitz equivalence, so it leads to the exact same growth rates within $\mathfrak{G}$. Since $\|\cdot\|\leq |\cdot|$, we always have

\[ \big[\|\phi^n(g)\|\big]\preceq \big[|\varphi^n(g)|\big] .\]

Moreover, $\big[|\varphi^n(g)|\big]\sim [1]$ holds if and only if a power of $\varphi$ fixes $g$, and $\big[\|\phi^n(g)\|\big]\sim[1]$ holds if and only if a power of $\phi$ preserves the conjugacy class of $g$. 

Given two growth rates $[a_n],[b_n]\in\mathfrak{G}$, the sum $[a_n]+[b_n]:=[a_n+b_n]$ is a well-defined growth rate. For $k\in\mathbb{N}$, we define $k\ast[a_n]:=[a_{kn}]$, that is, the equivalence class of the sequence $n\mapsto a_{kn}$. Similarly, we set $\frac{1}{k}\ast[a_n]:=[a_{\lfloor\frac{n}{k}\rfloor}]$.

For any finitely generated group $G$ and any $\psi\in\mathrm{Out}(G)$, we have the following identities

 \[   \overline{\mathfrak{o}}_{\rm top}(\psi^k)\sim k\ast\overline{\mathfrak{o}}_{\rm top}(\psi), \] \[\overline{\mathfrak{o}}_{\rm top}(\psi)\sim\tfrac{1}{k}\ast\overline{\mathfrak{o}}_{\rm top}(\psi^k) \]

for all $k\in\mathbb{N}$. Analogous identities hold for elements of $\mathrm{Aut}(G)$.

Recall the following:

Remark 2:  If a subgroup $\mathcal{O}\leq\mathrm{Out}(G)$ preserves the $G$--conjugacy class of a subgroup $H\leq G$, we can define a restriction $\mathcal{O}|_H\leq\mathrm{Out}(H)$: one considers the subgroup $U\leq\mathrm{Aut}(G)$ formed by automorphisms $\varphi$ with $\varphi(H)=H$ and outer class in $\mathcal{O}$, then one defines $\mathcal{O}|_H$ as the image of the composition $U\rightarrow \mathrm{Aut}(H)\rightarrow \mathrm{Out}(H)$ given by first restricting to $H$ and then projecting to outer automorphisms. The conjugation action $N_G(H)\curvearrowright H$ also determines a subgroup $C^G_H\leq\mathrm{Out}(H)$, and we always have $C^G_H\lhd\mathcal{O}|_H$. For each $\phi\in\mathcal{O}$, the possible restrictions of $\phi$ to $H$ form a canonical coset $\phi|_H\cdot C^G_H$ within $\mathcal{O}|_H$.

Remark 3:  Let $H\leq G$ be a convex-cocompact subgroup, and let $\phi\in\mathrm{Out}(G)$ preserve the $G$--conjugacy class of $H$. Although the restriction $\phi|_H\in\mathrm{Out}(H)$ is not uniquely defined in general (see above), the growth rate $\overline{\mathfrak{o}}_{\mathrm top}(\phi|_H)$ is well-defined. Indeed, any two possible restrictions $\phi|_H$ differ by the restriction to $H$ of an inner automorphism of $G$, and so conjugacy lengths grow at the same speed under their powers. (Recall that, since $H$ is convex-cocompact, it does not matter whether we compute conjugacy lengths with respect to a finite generating set of $H$ or $G$.)

Let $G$ be a special group. We now work to connect the enhanced JSJ decomposition of $G$ to the speed of growth of its outer automorphisms.

Let $\mathrm{Out}^0(G)\leq\mathrm{Out}(G)$ be the finite-index subgroup that preserves each $G$--conjugacy class in $\mathcal{S}(G)$. This subgroup has indeed finite index because $\mathcal{S}(G)$ is $\mathrm{Out}(G)$--invariant and consists of finitely many conjugacy classes of subgroups. 

When studying the top growth rate $\overline{\mathfrak{o}}_{\mathrm top}(\phi)$ of an outer automorphism $\phi\in\mathrm{Out}(G)$, it is often possible to reduce the problem to understanding the top growth rates of the restrictions of $\phi$ to the singular subgroups of $G$. For this reason, we introduce the following auxiliary growth rate.

Definition: For $\phi\in\mathrm{Out}^0(G)$, the singular growth rate of $\phi$ is

    \[ \overline{\mathfrak{o}}_{\mathrm sing}(\phi):=\sum_S \overline{\mathfrak{o}}_{\mathrm top}(\phi|_S), \]

  where the sum is taken over finitely many representatives $S\in\mathcal{S}(G)$ of the $G$--conjugacy classes of singular subgroups. For a general element $\phi\in\mathrm{Out}(G)$, we set $\overline{\mathfrak{o}}_{\mathrm sing}(\phi):=\frac{1}{k}\ast\overline{\mathfrak{o}}_{\mathrm top}(\phi^k)$ for any integer $k\in\mathbb{N}$ such that $\phi^k\in\mathrm{Out}^0(G)$.

The key result in linking growth of automorphisms with the JSJ is the following:

 Proposition 4: Let $G$ be special and $1$--ended. For any $\phi\in\mathrm{Out}(G)$, there exists a $\phi$--invariant $(\mathcal{ZZ}(G),\mathcal{K}_{\mathrm sing}(\phi))$--tree $G\curvearrowright T$ such that the $G$--stabiliser of each vertex of $T$ is:

(a) either an optimal quadratically hanging subgroup relative to $\mathcal{K}_{\mathrm sing}(\phi)$;

(b) or a convex-cocompact root-closed subgroup of $G$ that lies in $\mathcal{K}_{\mathrm sing}(\phi)$.

    Moreover, each edge $e\subseteq T$ with $G$--stabiliser not in $\mathcal{Z}(G)$ is incident to a type~(a) vertex.

The attentive reader who has been following this should note the extreme similarity with the main result from the previous post. The difference is that in the second case instead of saying 'rigid' we now conclude 'lies in $\mathcal{K}_{\mathrm sing}(\phi)$'. Indeed, this will follow from the result of the previous post and the following:

Lemma 5: Suppose that $G$ is $(\mathcal{Z}(G),\mathcal{H})$--rigid, for a collection of subgroups $\mathcal{H}\supseteq\mathcal{S}(G)$. Consider an outer automorphism $\phi\in\mathrm{Out}(G)$ and suppose that there exists an abstract growth rate $\mathfrak{o}\in\mathfrak{G}$ such that all subgroups in $\mathcal{H}$ are $\mathfrak{o}$--controlled. Then, we have $\overline{\mathfrak{o}}_{\mathrm top}(\phi)\preceq\mathfrak{o}$.

Proof of proposition 4: Theorem 1 of the previous post gives a $\phi$--invariant $(\mathcal{ZZ}(G),\mathcal{K}_{\mathrm sing}(\phi))$--tree $G\curvearrowright T$ such that each vertex group is either of type~(a), or it is a convex-cocompact root-closed subgroup $V\leq G$ that is $(\mathcal{Z}(V),\mathcal{K}_{\mathrm sing}(\phi)|_V)$--rigid in itself. Moreover, every edge of $T$ with $G$--stabiliser not in $\mathcal{Z}(G)$ is incident to a type~(a) vertex. We wish to show that $T$ is the tree we are looking for, which simply amounts to showing that all vertex groups not of type~(a) lie in $\mathcal{K}_{\mathrm sing}(\phi)$.

Thus, consider a convex-cocompact vertex group $V\leq G$ that is $(\mathcal{Z}(V),\mathcal{K}_{\mathrm sing}(\phi)|_V)$--rigid in itself. Note that, for every subgroup $H\in\mathcal{K}_{\mathrm sing}(\phi)|_V$ and every element $h\in H$, we have $\|\phi^n(h)\|\preceq\overline{\mathfrak{o}}_{\mathrm sing}(\phi)$, by definition (it does not matter whether we compute conjugacy lengths with respect to finite generating sets of $V$ or $G$, as $V$ is convex-cocompact). Up to raising $\phi$ to a power, we can assume that it preserves the $G$--conjugacy class of $V$, since $T$ is $\phi$--invariant. Now, Lemma 3 shows that $\overline{\mathfrak{o}}_{\mathrm top}(\phi|_V)\preceq\overline{\mathfrak{o}}_{\mathrm sing}(\phi)$, and hence $V\in\mathcal{K}_{\mathrm sing}(\phi)$ as required. $\blacksquare$