PolExp growth for automorphisms of toral relatively hyperbolic groups

TL;DR

The paper establishes that automorphisms of toral relatively hyperbolic groups exhibit PolExp growth: for any element or conjugacy class under iteration, the length grows like $n^d\λ^n$ with algebraic-integer $\λ$ and finite growth spectrum. The authors develop a robust geometric framework, including a metric Scott–Wall space and a refined JSJ decomposition for one-ended groups, to transfer local growth data from vertex groups to the whole group via a combination theorem. They also extend the theory to infinitely-ended groups using CTs and free-product techniques, and provide detailed analyses of palangres to link algebraic and geometric growth. The results yield precise growth types and spectra, with specialized refinements in hyperbolic and surface-type settings and implications for growth hierarchies and canonical subgroups. Overall, the work unifies algebraic and geometric perspectives to classify automorphism growth in this broad class of groups.

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 $n^dλ^n$ for some $d\in\mathbb{N}$ and some algebraic integer $λ\geq 1$. For a given $\varphi$, only finitely many values of $d$ and $λ$ 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 $n^dλ^n$ 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 $n^dλ^n$ if and only if $g$ is conjugate into one of the subgroups $K_i$.

Theorems & Definitions (27)

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