## Geometry & Topology

### Weighted $L^2$–cohomology of Coxeter groups

#### 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.

#### Article information

Source
Geom. Topol., Volume 11, Number 1 (2007), 47-138.

Dates
Received: 6 December 2006
Accepted: 6 January 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799832

Digital Object Identifier
doi:10.2140/gt.2007.11.47

Mathematical Reviews number (MathSciNet)
MR2287919

Zentralblatt MATH identifier
1173.20029

#### Citation

Davis, Michael W; Dymara, Jan; Januszkiewicz, Tadeusz; Okun, Boris. Weighted $L^2$–cohomology of Coxeter groups. Geom. Topol. 11 (2007), no. 1, 47--138. doi:10.2140/gt.2007.11.47. https://projecteuclid.org/euclid.gt/1513799832

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