Homological stability for families of Coxeter groups

We prove that certain families of Coxeter groups and inclusions
W1 ,→ W2 ,→ · · · satisfy homological stability, meaning that in each degree
the homology H∗(BWn) 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 Wn-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.