Geometry & Topology
- Geom. Topol.
- Volume 11, Number 1 (2007), 47-138.
Weighted $L^2$–cohomology of Coxeter groups
Given a Coxeter system and a positive real multiparameter , we study the “weighted –cohomology groups,” of a certain simplicial complex associated to . These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to and the multiparameter . They have a “von Neumann dimension” with respect to the associated “Hecke–von Neumann algebra” . The dimension of the –th cohomology group is denoted . It is a nonnegative real number which varies continuously with . When is integral, the are the usual –Betti numbers of buildings of type and thickness . For a certain range of , we calculate these cohomology groups as modules over and obtain explicit formulas for the . The range of for which our calculations are valid depends on the region of convergence of the growth series of . Within this range, we also prove a Decomposition Theorem for , analogous to a theorem of L Solomon on the decomposition of the group algebra of a finite Coxeter group.
Geom. Topol., Volume 11, Number 1 (2007), 47-138.
Received: 6 December 2006
Accepted: 6 January 2007
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 20C08: Hecke algebras and their representations 20E42: Groups with a $BN$-pair; buildings [See also 51E24] 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20J06: Cohomology of groups 46L10: General theory of von Neumann algebras 51E24: Buildings and the geometry of diagrams 57M07: Topological methods in group theory 58J22: Exotic index theories [See also 19K56, 46L05, 46L10, 46L80, 46M20]
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