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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.

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
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