#### Volume 11, issue 1 (2007)

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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
 arXiv: math.GT/0402377
##### Abstract

Given a Coxeter system $\left(W,S\right)$ and a positive real multiparameter $q$, we study the “weighted ${L}^{2}$–cohomology groups,” of a certain simplicial complex $\Sigma$ associated to $\left(W,S\right)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $\left(W,S\right)$ and the multiparameter $q$. They have a “von Neumann dimension” with respect to the associated “Hecke–von Neumann algebra” ${\mathsc{N}}_{q}$. The dimension of the $i$–th cohomology group is denoted ${b}_{q\left(\Sigma \right)}^{i}$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the ${b}_{q\left(\Sigma \right)}^{i}$ are the usual ${L}^{2}$–Betti numbers of buildings of type $\left(W,S\right)$ and thickness $q$. For a certain range of $q$, we calculate these cohomology groups as modules over ${\mathsc{N}}_{q}$ and obtain explicit formulas for the ${b}_{q\left(\Sigma \right)}^{i}$. 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 ${\mathsc{N}}_{q}$, 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