Homological stability for families of Coxeter groups

Abstract

We prove that certain families of Coxeter groups and inclusions $W_1\hookrightarrow W_2\hookrightarrow \cdots$ 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.