Product set growth in virtual subgroups of mapping class 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 $|U|⩾(\alpha |U|{)}^{\beta n}$, so long as no finite index subgroup of $⟨U⟩$ 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.

arXiv comment44 pages, 1 figure. Several of the main theorems have been upgraded. Corrections have been made in response to referee comments

References in corpus (9)

• Geometry of the Complex of Curves I: Hyperbolicity
• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Embedability between right-angled Artin groups
• Uniform hyperbolicity of the curve graph via surgery sequences
• Geometry of pseudocharacters
• Product set growth in groups and hyperbolic geometry
• Acylindrical group actions on quasi-trees
• The rates of growth in a hyperbolic group
• Product set growth in Burnside groups