Homological stability for classical groups

TL;DR

This paper develops a spectral sequence approach to prove homological stability for classical groups like unitary, special linear, and orthogonal groups over various fields, expanding understanding of their algebraic properties.

Contribution

It introduces a new spectral sequence framework based on Tits systems to establish homological stability for several classical groups over different fields.

Findings

Homological stability established for unitary groups over division rings.

Proved stability for special linear and orthogonal groups over infinite fields.

Spectral sequence method offers a unified approach to classical groups' homology.

Abstract

Associated to every group with a weak spherical Tits system of rank n+1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi subgroups frequently split as semidirect products of smaller groups, we prove homological stability results for unitary groups over division rings with infinite centre as well as for special linear and special orthogonal groups over infinite fields.