#### Volume 10, issue 2 (2006)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Other MSP Journals
Thin buildings

### Jan Dymara

Geometry & Topology 10 (2006) 667–694
 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
##### 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