Stability in the homology of congruence subgroups

TL;DR

This paper proves a strong form of representation stability called central stability for the homology groups of congruence subgroups of general linear groups over rings, extending classical stability results.

Contribution

It introduces a new concept of central stability for homology groups and demonstrates its implications for representation stability in congruence subgroups.

Findings

Homology groups of congruence subgroups satisfy central stability.

Central stability implies classical representation stability.

A new method analogous to classical stability machines is developed.

Abstract

The homology groups of many natural sequences of groups $\{G_n\}_{n=1}^{\infty}$ (e.g. general linear groups, mapping class groups, etc.) stabilize as $n \rightarrow \infty$. Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequences of groups whose homology groups do not stabilize in the classical sense actually stabilize in some sense as representations. They called this phenomena representation stability. We prove that the homology groups of congruence subgroups of $GL_n(R)$ (for almost any reasonable ring $R$) satisfy a strong version of representation stability that we call central stability. The definition of central stability is very different from Church-Farb's definition of representation stability (it is defined via a universal property), but we prove that it implies representation stability. Our main tool is a new machine for proving central stability that is analogous to the classical homological stability machine.