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ISSN (electronic): 1472-2739
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The $L^2$–(co)homology of groups with hierarchies

Boris Okun and Kevin Schreve

Algebraic & Geometric Topology 16 (2016) 2549–2569

We study group actions on manifolds that admit hierarchies, which generalizes the idea of Haken n–manifolds introduced by Foozwell and Rubinstein. We show that these manifolds satisfy the Singer conjecture in dimensions n 4. 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.

Singer conjecture, Haken $n$–manifolds, aspherical manifolds, Coxeter groups, action dimension, hierarchy
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20J05
Received: 5 July 2014
Revised: 24 September 2015
Accepted: 6 April 2016
Published: 7 November 2016
Boris Okun
Department of Mathematical Sciences
University of Wisconsin–Milwaukee
PO Box 413
Milwaukee, WI 53201-0413
United States
Kevin Schreve
Department of Mathematics
University of Michigan
530 Church St.
Ann Arbor, MI 48109
United States