Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 16, Number 5 (2016), 2549-2569.
The $L^2$–(co)homology of groups with hierarchies
We study group actions on manifolds that admit hierarchies, which generalizes the idea of Haken –manifolds introduced by Foozwell and Rubinstein. We show that these manifolds satisfy the Singer conjecture in dimensions . Our main application is to Coxeter groups whose Davis complexes are manifolds; we show that the natural action of these groups on the Davis complex has a hierarchy. Our second result is that the Singer conjecture is equivalent to the cocompact action dimension conjecture, which is a statement about all groups, not just fundamental groups of closed aspherical manifolds.
Algebr. Geom. Topol., Volume 16, Number 5 (2016), 2549-2569.
Received: 5 July 2014
Revised: 24 September 2015
Accepted: 6 April 2016
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20J05: Homological methods in group theory
Okun, Boris; Schreve, Kevin. The $L^2$–(co)homology of groups with hierarchies. Algebr. Geom. Topol. 16 (2016), no. 5, 2549--2569. doi:10.2140/agt.2016.16.2549. https://projecteuclid.org/euclid.agt/1510841220