The L2–(co)homology of groups with hierarchies

Okun, Boris¹ ; Schreve, Kevin²

¹Department of Mathematical Sciences, University of Wisconsin–Milwaukee, PO Box 413, Milwaukee, WI 53201-0413, United States
²Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109, United States

Algebraic and Geometric Topology, Tome 16 (2016) no. 5, pp. 2549-2569

Résumé

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 ≤ 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.

Détail

DOI : 10.2140/agt.2016.16.2549

Classification : 20F65, 20J05
Keywords: Singer conjecture, Haken $n$–manifolds, aspherical manifolds, Coxeter groups, action dimension, hierarchy

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