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 $n\le 4$. 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.

11 pages

References in corpus (4)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Euler characteristics of generalized Haken manifolds
• The universal covering space of a Haken $n$--manifold
• The Euler Characteristic of a Haken 4-Manifold

Cited by in corpus (4)

• Mod $p$ and torsion homology growth in nonpositive curvature
• Survey on L^2-invariants and 3-manifolds
• The weighted Singer conjecture for Coxeter groups in dimensions three and four
• Coxeter Groups, Ruins, and Weighted $L^2$-cohomology