Weighted L2–cohomology of Coxeter groups

Davis, Michael W; Dymara, Jan; Januszkiewicz, Tadeusz; Okun, Boris

doi:10.2140/gt.2007.11.47

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Weighted $L^2$–cohomology of Coxeter groups

Michael W Davis, Jan Dymara, Tadeusz Januszkiewicz and Boris Okun

Geometry & Topology 11 (2007) 47–138

DOI: 10.2140/gt.2007.11.47

arXiv: math.GT/0402377

Abstract

Given a Coxeter system $(W, S)$ and a positive real multiparameter $q$ , we study the “weighted $L^{2}$ –cohomology groups,” of a certain simplicial complex $Σ$ associated to $(W, S)$ . These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $(W, S)$ and the multiparameter $q$ . They have a “von Neumann dimension” with respect to the associated “Hecke–von Neumann algebra” $N_{q}$ . The dimension of the $i$ –th cohomology group is denoted $b_{q (Σ)}^{i}$ . It is a nonnegative real number which varies continuously with $q$ . When $q$ is integral, the $b_{q (Σ)}^{i}$ are the usual $L^{2}$ –Betti numbers of buildings of type $(W, S)$ and thickness $q$ . For a certain range of $q$ , we calculate these cohomology groups as modules over $N_{q}$ and obtain explicit formulas for the $b_{q (Σ)}^{i}$ . The range of $q$ for which our calculations are valid depends on the region of convergence of the growth series of $W$ . Within this range, we also prove a Decomposition Theorem for $N_{q}$ , analogous to a theorem of L Solomon on the decomposition of the group algebra of a finite Coxeter group.

Keywords

Coxeter group, Hecke algebra, von Neumann algebra, building, $L^2$–cohomology

Mathematical Subject Classification 2000

Primary: 20F55

Secondary: 20C08, 20E42, 20F65, 20J06, 46L10, 51E24, 57M07, 58J22

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Publication

Received: 6 December 2006

Accepted: 6 January 2007

Published: 24 February 2007

Proposed: Wolfgang Lueck

Seconded: Steve Ferry, Martin Bridson

Authors

Michael W Davis
The Ohio State University Department of Mathematics 231 W 18th Ave Columbus, Ohio 43210–1174 United States

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

Tadeusz Januszkiewicz
The Ohio State University Department of Mathematics 231 W 18th Ave Columbus, Ohio 43210–1174 United States	Instytut Matematyczny Polskiej Akademii Nauk

Boris Okun
University of Wisconsin–Milwaukee Department of Mathematical Sciences PO Box 413 Milwaukee WI 53201–0413 United States

Volume 11, issue 1 (2007)