Volume 11, issue 1 (2007)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Weighted $L^2$–cohomology of Coxeter groups

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

Geometry & Topology 11 (2007) 47–138

arXiv: math.GT/0402377

Abstract

Given a Coxeter system (W,S) and a positive real multiparameter q, we study the “weighted L2–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” Nq. The dimension of the i–th cohomology group is denoted bq(Σ)i. It is a nonnegative real number which varies continuously with q. When q is integral, the bq(Σ)i are the usual L2–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 Nq and obtain explicit formulas for the bq(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 Nq, 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