L²-homology and reciprocity for right-angled Coxeter groups

• Autorzy: Boris Okun , Richard Scott
• Języki publikacji: EN

Abstrakty

EN

Let W be a Coxeter group and let μ be an inner product on the group algebra ℝW. We say that μ is admissible if it satisfies the axioms for a Hilbert algebra structure. Any such inner product gives rise to a von Neumann algebra $𝒩_{μ}$ containing ℝW. Using these algebras and the corresponding von Neumann dimensions we define $L²_{μ}$-Betti numbers and an $L²_{μ}$-Euler charactersitic for W. We show that if the Davis complex for W is a generalized homology manifold, then these Betti numbers satisfy a version of Poincaré duality. For arbitrary Coxeter groups, finding interesting admissible products is difficult; however, if W is right-angled, there are many. We exploit this fact by showing that when W is right-angled, there exists an admissible inner product μ such that the $L²_{μ}$-Euler characteristic is 1/W(t) where W(t) is the growth series corresponding to a certain normal form for W. We then show that a reciprocity formula for this growth series that was recently discovered by the second author is a consequence of Poincaré duality.

Kategorie tematyczne

• 20F65: Geometric group theory
• 20F55: Reflection and Coxeter groups
• 20C08: Hecke algebras and their representations
• 20J06: Cohomology of groups
• 46L10: General theory of von Neumann algebras
• 57M07: Topological methods in group theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2011
• Tom: 214
• Numer: 1
• Strony: 27-56

Daty

wydano

2011

Twórcy

autor

Boris Okun

• Department of Mathematical Sciences, University of Wisconsin, Milwaukee, WI 53201, U.S.A.

autor

Richard Scott

• Department of Mathematics, and Computer Science, Santa Clara University, Santa Clara, CA 95053, U.S.A.

Identyfikatory

DOI

10.4064/fm214-1-3