## $L^2$-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.