#### Volume 16, issue 5 (2016)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
The $L^2$–(co)homology of groups with hierarchies

### Boris Okun and Kevin Schreve

Algebraic & Geometric Topology 16 (2016) 2549–2569
##### 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 $n\le 4$. 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.

##### Keywords
Singer conjecture, Haken $n$–manifolds, aspherical manifolds, Coxeter groups, action dimension, hierarchy
Primary: 20F65
Secondary: 20J05