Algebraic & Geometric Topology

The $L^2$–(co)homology of groups with hierarchies

Boris Okun and Kevin Schreve

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Abstract

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.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 5 (2016), 2549-2569.

Dates
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
https://projecteuclid.org/euclid.agt/1510841220

Digital Object Identifier
doi:10.2140/agt.2016.16.2549

Mathematical Reviews number (MathSciNet)
MR3572340

Zentralblatt MATH identifier
1383.20028

Subjects
Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20J05: Homological methods in group theory

Keywords
Singer conjecture Haken $n$–manifolds aspherical manifolds Coxeter groups action dimension hierarchy

Citation

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


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