Geometry & Topology

Weighted $L^2$–cohomology of Coxeter groups

Michael W Davis, Jan Dymara, Tadeusz Januszkiewicz, and Boris Okun

Full-text: Open access

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

Subjects
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]

Keywords
Coxeter group Hecke algebra von Neumann algebra building $L^2$–cohomology

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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