# Central stability for the homology of congruence subgroups and the second homology of Torelli groups

## TL;DR

This paper establishes new stability results for the second homology of Torelli groups and certain congruence subgroups, advancing understanding of their algebraic structures and homological properties.

## Contribution

It introduces a general theorem on syzygies of modules with finite polynomial degree, strengthening previous stability results for these groups.

## Key findings

- Proves stability for second homology of Torelli groups.
- Establishes stability for homology of certain congruence subgroups.
- Improves upon previous work by Putman-Sam.

## Abstract

We prove a representation stability result for the second homology groups of Torelli subgroups of mapping class groups and automorphism groups of free groups. This strengthens the results of Boldsen-Hauge Dollerup and Day-Putman. We also prove a new representation stability result for the homology of certain congruence subgroups, partially improving upon the work of Putman-Sam. These results follow from a general theorem on syzygies of certain modules with finite polynomial degree.