×

zbMATH — the first resource for mathematics

The \(L^2\)-(co)homology of groups with hierarchies. (English) Zbl 1383.20028
Summary: We study group actions on manifolds that admit hierarchies, which generalizes the idea of Haken \(n\)-manifolds introduced by B. Foozwell and H. Rubinstein [Contemp. Math. 560, 71–84 (2011; Zbl 1335.57038)]. We show that these manifolds satisfy the Singer conjecture in dimensions \(n \leq 4\). 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.

MSC:
20F65 Geometric group theory
20J05 Homological methods in group theory
20F55 Reflection and Coxeter groups (group-theoretic aspects)
57M07 Topological methods in group theory
57M60 Group actions on manifolds and cell complexes in low dimensions
57S25 Groups acting on specific manifolds
PDF BibTeX XML Cite
Full Text: DOI arXiv