Homological stability for families of Coxeter groups

Hepworth, Richard

doi:10.2140/agt.2016.16.2779

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Homological stability for families of Coxeter groups

Richard Hepworth

Algebraic & Geometric Topology 16 (2016) 2779–2811

DOI: 10.2140/agt.2016.16.2779

Abstract

We prove that certain families of Coxeter groups and inclusions $W_{1} ↪ W_{2} ↪ \dots$ satisfy homological stability, meaning that in each degree the homology $H_{*} (B W_{n})$ 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

Mathematical Subject Classification 2010

Primary: 20F55

Secondary: 20J06

References

Publication

Received: 19 February 2015

Revised: 23 December 2015

Accepted: 12 January 2016

Published: 7 November 2016

Authors

Richard Hepworth
Institute of Mathematics University of Aberdeen Aberdeen AB24 3UE United Kingdom
http://www.abdn.ac.uk/ncs/profiles/r.hepworth/

Volume 16, issue 5 (2016)