Improved stability ranges in the homology of Torelli and congruence subgroups

TL;DR

This paper enhances the known stability ranges in the homology of Torelli and congruence subgroups, providing improved bounds for their algebraic and topological properties as group representations.

Contribution

It introduces improved stability bounds for the second homology of Torelli groups and for higher homology of congruence subgroups, advancing understanding of their algebraic structures.

Findings

Extended stability ranges for H2(Torelli groups) as GL_n(Z)-representations

Extended stability ranges for H2(Torelli groups) as Sp_2g(Z)-representations

Extended stability ranges for Hk(congruence subgroups) as SL_n^U(R/I)-representations

Abstract

We improve the central stability ranges for H2(Torelli subgroup of Aut(Fn)'s) as GL_n(Z)-representations, H2(Torelli subgroup of mapping class groups) as Sp_2g(Z)-representations, Hk(congruence subgroups of GL_n(R)'s) as SL_n^U(R/I)-representations.