PolExp growth for automorphisms of toral relatively hyperbolic groups

TL;DR

This paper proves that the conjugacy and word lengths of elements in toral relatively hyperbolic groups grow like a polynomial times exponential function under automorphisms, and characterizes polynomial growth subgroups in hyperbolic groups.

Contribution

It establishes precise growth rates for automorphisms of toral relatively hyperbolic groups and generalizes polynomial subgroup concepts for hyperbolic groups.

Findings

Growth of conjugacy length is like n^d λ^n for some d and λ

Only finitely many growth parameters occur for each automorphism

Polynomial growth subgroups are characterized by malnormal quasiconvex subgroups

Abstract

Let $G$ be a toral relatively hyperbolic group, and let $\varphi\in\mathrm{Aut}(G)$. We prove that, under iteration of $\varphi$, the conjugacy length $||\varphi^n(g)||$ of every element $g\in G$ grows like $n^d\lambda^n$ for some $d\in\mathbb{N}$ and some algebraic integer $\lambda\geq 1$. For a given $\varphi$, only finitely many values of $d$ and $\lambda$ occur as $g$ varies in $G$. The same statements hold for the growth of the word length $|\varphi^n(g)|$. For $G$ hyperbolic, we generalize polynomial subgroups: we show that, for a given growth type $n^d\lambda^n$ other than $1$, there is a malnormal family of quasiconvex subgroups $K_1,\dots,K_p$ such that a conjugacy class $[g]$ grows at most like $n^d\lambda^n$ if and only if $g$ is conjugate into one of the subgroups $K_i$.