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

TL;DR

This paper investigates the $L^2$-(co)homology of groups acting on manifolds with hierarchies, proving the Singer conjecture in low dimensions and establishing equivalence with the cocompact action dimension conjecture.

Contribution

It introduces the concept of hierarchies in group actions on manifolds, extends the Singer conjecture to these cases, and links it to the broader cocompact action dimension conjecture.

Findings

Manifolds with hierarchies satisfy the Singer conjecture in dimensions up to 4.

Coxeter groups with Davis complexes as manifolds admit hierarchical actions.

The Singer conjecture is equivalent to the cocompact action dimension conjecture.

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.