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 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” N_q. The dimension of the i–th cohomology group is denoted bⁱ_q(Σ). It is a nonnegative real number which varies continuously with q. When q is integral, the bⁱ_q(Σ) 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 N_q and obtain explicit formulas for the bⁱ_q(Σ). 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, L²–cohomology

Mathematical Subject Classification

Primary: 20F55

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

References

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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