$L^2$-homology and reciprocity for right-angled Coxeter groups

Boris Okun ¹   ; Richard Scott ²

¹ Department of Mathematical Sciences University of Wisconsin Milwaukee, WI 53201, U.S.A.
² Department of Mathematics and Computer Science Santa Clara University Santa Clara, CA 95053, U.S.A.

Fundamenta Mathematicae, Tome 214 (2011) no. 1, pp. 27-56

Résumé

Let $W$ be a Coxeter group and let $\mu$ be an inner product on the group algebra $\mathbb R W$. We say that $\mu$ is admissible if it satisfies the axioms for a Hilbert algebra structure. Any such inner product gives rise to a von Neumann algebra $\mathcal N_{\mu}$ containing $\mathbb R W$. Using these algebras and the corresponding von Neumann dimensions we define $L^2_{\mu}$-Betti numbers and an $L^2_{\mu}$-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 $\mu$ such that the $L^2_{\mu}$-Euler characteristic is $1/W(\mathbf{t})$ where $W(\mathbf{t})$ 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.

Détail

DOI : 10.4064/fm214-1-3

Keywords: coxeter group inner product group algebra mathbb say admissible satisfies axioms hilbert algebra structure inner product gives rise von neumann algebra mathcal containing mathbb using these algebras corresponding von neumann dimensions define betti numbers euler charactersitic davis complex generalized homology manifold these betti numbers satisfy version poincar duality arbitrary coxeter groups finding interesting admissible products difficult however right angled there many exploit showing right angled there exists admissible inner product euler characteristic mathbf where mathbf growth series corresponding certain normal form reciprocity formula growth series recently discovered second author consequence poincar duality