Commensurability of geometric subgroups of mapping class groups

TL;DR

This paper investigates the relationships between geometric subgroups of mapping class groups, characterizing when they are commensurable or virtually abelian, and explores implications for unitary representations.

Contribution

It provides a comprehensive characterization of commensurability among geometric subgroups and describes their commensurators, advancing understanding of mapping class group structures.

Findings

Characterization of virtually abelian geometric subgroups

Algebraic and geometric criteria for subgroup commensurability

Description of the commensurator in terms of subsurface stabilizers

Abstract

Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly disconnected). We characterise the subsurfaces which lead to virtually abelian geometric subgroups. We provide algebraic and geometric conditions under which two geometric subgroups are commensurable. We also describe the commensurator of a geometric subgroup in terms of the stabiliser of the underlying subsurface. Finally, we show some applications of our analysis to the theory of irreducible unitary representations of mapping class groups.