English

PolExp growth for automorphisms of toral relatively hyperbolic groups

Group Theory 2025-12-08 v1

Abstract

Let GG be a toral relatively hyperbolic group, and let φAut(G)\varphi\in\mathrm{Aut}(G). We prove that, under iteration of φ\varphi, the conjugacy length φn(g)||\varphi^n(g)|| of every element gGg\in G grows like ndλnn^d\lambda^n for some dNd\in\mathbb{N} and some algebraic integer λ1\lambda\geq 1. For a given φ\varphi, only finitely many values of dd and λ\lambda occur as gg varies in GG. The same statements hold for the growth of the word length φn(g)|\varphi^n(g)|. For GG hyperbolic, we generalize polynomial subgroups: we show that, for a given growth type ndλnn^d\lambda^n other than 11, there is a malnormal family of quasiconvex subgroups K1,,KpK_1,\dots,K_p such that a conjugacy class [g][g] grows at most like ndλnn^d\lambda^n if and only if gg is conjugate into one of the subgroups KiK_i.

Keywords

Cite

@article{arxiv.2512.05569,
  title  = {PolExp growth for automorphisms of toral relatively hyperbolic groups},
  author = {Rémi Coulon and Arnaud Hilion and Camille Horbez and Gilbert Levitt},
  journal= {arXiv preprint arXiv:2512.05569},
  year   = {2025}
}