# Partial Torelli groups and homological stability

## TL;DR

This paper establishes homological stability results for certain subgroups of the mapping class group, extending known theorems to new contexts involving fixed homology data and finite group actions.

## Contribution

It introduces new homological stability theorems for subgroups of the mapping class group related to fixed homology and finite group maps, generalizing prior work on braid groups.

## Key findings

- Proves stability for subgroups fixing part of the homology.
- Establishes stability for subgroups preserving a map to a finite group.
- Uses complexes of subsurfaces to generalize previous results.

## Abstract

We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, which can be viewed as a mapping class group version of a theorem of Ellenberg-Venkatesh-Westerland about braid groups. These results require studying various simplicial complexes formed by subsurfaces of the surface, generalizing work of Hatcher-Vogtmann.