Product set growth in virtual subgroups of mapping class groups

TL;DR

This paper investigates product set growth in groups acting acylindrically on hyperbolic spaces, revealing a growth dichotomy in mapping class groups and their virtual subgroups, with implications for right-angled Artin groups.

Contribution

It establishes a growth dichotomy for finitely generated subgroups of mapping class groups and extends results to virtual subgroups and right-angled Artin groups.

Findings

Existence of positive constants  and  for growth bounds

Dichotomy in subgroup growth behavior

Application to virtually special groups

Abstract

We study product set growth in groups with acylindrical actions on quasi-trees and, more generally, hyperbolic spaces. As a consequence, we show that for every surface $S$ of finite type, there exist $\alpha,\beta>0$ such that for any finite symmetric subset $U$ of the mapping class group $MCG(S)$ we have $|U|\geqslant (\alpha|U|)^{\beta n}$, so long as no finite index subgroup of $\langle U\rangle$ has an infinite centre. This gives us a dichotomy for the finitely generated subgroups of mapping class groups, which extends to virtual subgroups. As right-angled Artin groups embed as subgroups of mapping class groups, this result applies to them, and so also applies to finitely generated virtually special groups. We separately prove that we can quickly generate loxodromic elements in right-angled Artin groups, which by a result of Fujiwara shows that the set of growth rates for many of their subgroups are well-ordered.