Thin buildings

Dymara, Jan

doi:10.2140/gt.2006.10.667

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Thin buildings

Jan Dymara

Geometry & Topology 10 (2006) 667–694

DOI: 10.2140/gt.2006.10.667

arXiv: math.GT/0601005

Abstract

Let $X$ be a building of uniform thickness $q + 1$ . $L^{2}$ –Betti numbers of $X$ are reinterpreted as von-Neumann dimensions of weighted $L^{2}$ –cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness $q$ . The weighted cohomology makes sense for all real positive values of $q$ , and is computed for small $q$ . If the Davis complex of the Coxeter group is a manifold, a version of Poincaré duality allows to deduce that the $L^{2}$ –cohomology of a building with large thickness is concentrated in the top dimension.

Keywords

building, $L^2$-cohomology, Hecke algebra

Mathematical Subject Classification 2000

Primary: 20F55

Secondary: 20C08, 58J22, 20E42

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Publication

Received: 6 January 2006

Accepted: 30 April 2006

Published: 24 May 2006

Proposed: Wolfgang Lück

Seconded: Martin Bridson, Steve Ferry

Authors

Jan Dymara
Instytut Matematyczny Uniwersytet Wrocławski pl. Grunwaldzki 2/4 50-384 Wrocław Poland

Volume 10, issue 2 (2006)