L²-homology and reciprocity for right-angled Coxeter groups

Boris Okun, and Richard Scott. "L²-homology and reciprocity for right-angled Coxeter groups." Fundamenta Mathematicae 214.1 (2011): 27-56. <http://eudml.org/doc/282921>.

@article{BorisOkun2011,
abstract = {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 $_\{μ\}$ 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.},
author = {Boris Okun, Richard Scott},
journal = {Fundamenta Mathematicae},
keywords = {Coxeter groups; -cohomology; reciprocity; von Neumann algebras; Davis complexes; generalized homology manifolds; Betti numbers; Poincaré duality},
language = {eng},
number = {1},
pages = {27-56},
title = {L²-homology and reciprocity for right-angled Coxeter groups},
url = {http://eudml.org/doc/282921},
volume = {214},
year = {2011},
}

TY - JOUR
AU - Boris Okun
AU - Richard Scott
TI - L²-homology and reciprocity for right-angled Coxeter groups
JO - Fundamenta Mathematicae
PY - 2011
VL - 214
IS - 1
SP - 27
EP - 56
AB - 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 $_{μ}$ 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.
LA - eng
KW - Coxeter groups; -cohomology; reciprocity; von Neumann algebras; Davis complexes; generalized homology manifolds; Betti numbers; Poincaré duality
UR - http://eudml.org/doc/282921
ER -