The girth alternative for mapping class groups

TL;DR

This paper investigates the girth properties of subgroups within the mapping class group, establishing a dichotomy that such subgroups are either virtually abelian or have infinite girth, with specific conditions for non-cyclic groups.

Contribution

It introduces a girth alternative for subgroups of the mapping class group, characterizing their algebraic structure based on girth properties.

Findings

Subgroups are either virtually abelian or have infinite girth.

The dichotomy is mutually exclusive for non-infinite cyclic subgroups.

Provides a new perspective on subgroup structure in mapping class groups.

Abstract

The girth of a finitely generated group G is the supremum of the girth of Cayley graphs for G over all finite generating sets. Let G be a finitely generated subgroup of the mapping class group Mod(S), where S is a compact orientable surface. Then, either G is virtually abelian or it has infinite girth; moreover, if we assume that G is not infinite cyclic, these alternatives are mutually exclusive.