The rates of growth in an acylindrically hyperbolic group

Abstract

Let G be an acylindrically hyperbolic group on a δhyperbolic space X. Assume there exists M such that for any generating set S of G, S M contains a hyperbolic element on X. Suppose that G is equationally Noetherian. Then we show the set of the growth rate of G is well-ordered (Theorem 1.1).The conclusion is known for hyperbolic groups, and this is a generalization. Our result applies to all lattices in simple Lie groups of rank-1 (Theorem 1.2), and more generally, some family of relatively hyperbolic groups (Theorem 1.3). A potential application is a mapping class group, to which the theorem applies if it is equationally Noetherian.

Lower bound on growth of non-elementary subgroups in relatively hyperbolic groups

“…Set L :" 4pm 1 `1q ě 10. By (3), we have dpo, soq ď C 0 `dpo, boq for any s P S, and thus (10) dpy, syq ď p2m 1 `1qdpo, boq `C0 ă L 2 dpy, byq which yields for any c :" s ´1s 1 with s ‰ s 1 P S,…”

“…This question has been answered for equationally Noetherian relatively hyperbolic groups by Fujiwara's preprint[10] (posted on 2 Mar. 2021, one day earlier than ours on arXiv).…”