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 containing ℝW. Using these algebras and the corresponding von Neumann dimensions we define -Betti numbers and an -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 -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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-1-3,
author = {Boris Okun and Richard Scott},
title = {L$^2$-homology and reciprocity for right-angled Coxeter groups},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {27-56},
zbl = {1242.20050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-1-3}
}
Boris Okun; Richard Scott. L²-homology and reciprocity for right-angled Coxeter groups. Fundamenta Mathematicae, Tome 215 (2011) pp. 27-56. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-1-3/