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Homological stability for families of Coxeter groups

Richard Hepworth

Algebraic & Geometric Topology 16 (2016) 2779–2811

We prove that certain families of Coxeter groups and inclusions W1W2 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.

homological stability, Coxeter groups
Mathematical Subject Classification 2010
Primary: 20F55
Secondary: 20J06
Received: 19 February 2015
Revised: 23 December 2015
Accepted: 12 January 2016
Published: 7 November 2016
Richard Hepworth
Institute of Mathematics
University of Aberdeen
Aberdeen AB24 3UE
United Kingdom