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arXiv · 1407.1340

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

Boris Okun·Kevin Schreve
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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.

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Boris Okun, Kevin Schreve. 2014-07-04. The $L^2$-(co)homology of groups with hierarchies. https://doi.org/10.2140/agt.2016.16.2549

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