Product set growth in virtual subgroups of mapping class groups

Kerr, Alice

doi:10.2140/agt.2025.25.2757

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ISSN (electronic): 1472-2739

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Product set growth in virtual subgroups of mapping class groups

Alice Kerr

Algebraic & Geometric Topology 25 (2025) 2757–2806

DOI: 10.2140/agt.2025.25.2757

Abstract

We study product set growth in groups with acylindrical actions on quasitrees and, more generally, hyperbolic spaces. As a consequence, we show that for every surface $S$ of finite type, there exist $α, β > 0$ such that for any finite symmetric subset $U$ of the mapping class group $MCG (S)$ we have $| U^{n} | \geq {(α | U |)}^{β n}$ , so long as no finite-index subgroup of $⟨ U ⟩$ has 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

(Groups Geom. Dynam. 19 (2025) 109–167) shows that the set of growth rates for many of their subgroups are well ordered.

Keywords

group growth, acylindrically hyperbolic groups, mapping class groups, right-angled Artin groups

Mathematical Subject Classification

Primary: 20F36, 20F65, 20F67, 20F69

References

Publication

Received: 5 July 2021

Revised: 31 January 2024

Accepted: 23 July 2024

Published: 17 September 2025

Authors

Alice Kerr
School of Mathematics University of Birmingham Birmingham United Kingdom

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Volume 25, issue 5 (2025)