Equivariant group presentations and the second homology group of the Torelli group

TL;DR

This paper introduces a theory of equivariant group presentations and applies it to show that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,bZ)$-module, advancing understanding of its algebraic structure.

Contribution

It develops a new theory of equivariant group presentations and demonstrates its application to the second homology group of the Torelli group, showing finite generation as a module.

Findings

Second homology group of Torelli subgroup is finitely generated as an $Sp(2g,bZ)$-module

Introduces a new framework for equivariant group presentations

Links group homology to symplectic group actions

Abstract

We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,\mathbb{Z})$-module.