Abelian subgroups of the mapping class groups for non-orientable surfaces

TL;DR

This paper extends the understanding of abelian subgroups in mapping class groups to non-orientable surfaces, demonstrating finite generation and constructing isomorphic groups for torsion-free subgroups.

Contribution

It adapts existing proofs from orientable to non-orientable surfaces, establishing finite generation and explicit isomorphisms for certain subgroups.

Findings

Finitely generated abelian subgroups for non-orientable surfaces

Construction of isomorphic groups for torsion-free subgroups

Extension of known results from orientable to non-orientable cases

Abstract

Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free subgroup of the mapping class groups.