Weighted $L^2$-cohomology of Coxeter groups

Details Weighted $L^2$-cohomology of Coxeter groups Weighted $L^2$-cohomology of Coxeter groups

2004-02-23

arxiv.org/abs/math/0402377v2

Mathematics

Geometric Topology

minor changes

Scientific paper

Given a Coxeter system $(W,S)$ and a positive real multiparameter $\bq$, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $\Sigma$ 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 $b^i_q(\Sigma)$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b^i_q(\Sigma)$ 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 $b^i_q(\Sigma)$. 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.

Affiliated with

Also associated with

No associations

Rating

Weighted $L^2$-cohomology of Coxeter groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Weighted $L^2$-cohomology of Coxeter groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Weighted $L^2$-cohomology of Coxeter groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-352533