## Algebraic & Geometric Topology

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

#### 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
Revised: 24 September 2015
Accepted: 6 April 2016
First available in Project Euclid: 16 November 2017

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
Secondary: 20J05: Homological methods in group theory

#### 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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