Pré-Publication, Document De Travail Année : 2025

PolExp growth for automorphisms of toral relatively hyperbolic groups

Résumé

Let $G$ be a toral relatively hyperbolic group, and let $φ\in\mathrm{Aut}(G)$. We prove that, under iteration of $φ$, the conjugacy length $||φ^n(g)||$ 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 $φ$, 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 $|φ^n(g)|$. 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 $K_i$.

Dates et versions

hal-05403402 , version 1 (08-12-2025)

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Rémi Coulon, Arnaud Hilion, Camille Horbez, Gilbert Levitt. PolExp growth for automorphisms of toral relatively hyperbolic groups. 2025. ⟨hal-05403402⟩
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