Weighted $L^2$-cohomology of Coxeter groups

Abstract

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 $i^{t}h$ cohomology group is denoted $b_q^i(\Sigma )$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b_q^i(\Sigma )$ 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 $b_q^i(\Sigma )$. 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.

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References in corpus (3)

• Real Root Conjecture fails for five and higher dimensional spheres
• Thin buildings
• Cohomology of Coxeter groups with group ring coefficients: II

Cited by in corpus (19)

• Real Root Conjecture fails for five and higher dimensional spheres
• Thin buildings
• Arithmetic aspects of symmetric edge polytopes
• Factoriality of Hecke-von Neumann algebras of right-angled Coxeter groups
• Vanishing of ${\ell }^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence
• $L^2$-Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices
• Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras
• Random graph products of finite groups are rational duality groups
• The ${\ell }^2$-homology of even Coxeter groups
• On MASAs in $q$-deformed von Neumann algebras
• The Poincare series of the hyperbolic Coxeter groups with finite volume of fundamental domains
• The weighted Singer conjecture for Coxeter groups in dimensions three and four
• Equivariant triangulations of tori of compact Lie groups and hyperbolic extension to non-crystallographic Coxeter groups
• The flag upper bound theorem for 3- and 5-manifolds
• Weighted L^2-cohomology of Coxeter groups based on barycentric subdivisons
• A note on the rational homological dimension of lattices in positive characteristic
• Action Dimension of Lattices in Euclidean Buildings
• Eulerian cube complexes and reciprocity
• On the centre of Iwahori-Hecke algebras