Homological stability for families of Coxeter groups

TL;DR

This paper establishes homological stability for certain families of Coxeter groups, including new results for type D, by analyzing the connectivity of associated simplicial complexes using geometric group theory techniques.

Contribution

It provides a unified proof of homological stability for Coxeter groups of types A, B, and D, with a novel approach for type D using the basic construction.

Findings

Homological stability holds for Coxeter groups of types A, B, and D.

The connectivity of a certain simplicial complex is key to the proof.

New stability result is established for type D Coxeter groups.

Abstract

We prove that certain families of Coxeter groups and inclusions $W_1\hookrightarrow W_2\hookrightarrow...$ satisfy homological stability, meaning that in each degree the homology $H_\ast(BW_n)$ is eventually independent of $n$. This gives a uniform treatment of homological stability for the families of Coxeter groups of type $A_n$, $B_n$ and $D_n$, recovering existing results in the first two cases, and giving a new result in the third. The key step in our proof is to show that a certain simplicial complex with $W_n$-action is highly connected. To do this we show that the barycentric subdivision is an instance of the 'basic construction', and then use Davis's description of the basic construction as an increasing union of chambers to deduce the required connectivity.