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The $L^2$–(co)homology
of groups with hierarchies
Boris Okun and Kevin Schreve |
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Algebraic & Geometric Topology 16 (2016) 2549–2569
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Abstract |
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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. |
Keywords
Singer conjecture, Haken $n$–manifolds, aspherical
manifolds, Coxeter groups, action dimension, hierarchy
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Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20J05
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References |
Publication
Received: 5 July 2014
Revised: 24 September 2015
Accepted: 6 April 2016
Published: 7 November 2016
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Authors |
| Boris Okun | |
| Department of Mathematical
Sciences University of Wisconsin–Milwaukee PO Box 413 Milwaukee, WI 53201-0413 United States |
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| Kevin Schreve | |
| Department of Mathematics University of Michigan 530 Church St. Ann Arbor, MI 48109 United States |
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