$L^2$-homology and reciprocity for right-angled Coxeter groups
Volume 214 / 2011
Abstract
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.
