#### Volume 16, issue 5 (2016)

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Homological stability for families of Coxeter groups

### Richard Hepworth

Algebraic & Geometric Topology 16 (2016) 2779–2811
##### Abstract

We prove that certain families of Coxeter groups and inclusions ${W}_{1}↪{W}_{2}↪\cdots \phantom{\rule{0.3em}{0ex}}$ satisfy homological stability, meaning that in each degree the homology ${H}_{\ast }\left(B{W}_{n}\right)$ is eventually independent of $n$. This gives a uniform treatment of homological stability for the families of Coxeter groups of type $A$, $B$ and $D$, 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.

##### Keywords
homological stability, Coxeter groups
Primary: 20F55
Secondary: 20J06