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L²-homology and reciprocity for right-angled Coxeter groups

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Okun B., Scott R.

Fundamenta Mathematicae

2011

tom 214

nr 1

27-56

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 $𝒩_{μ}$ 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.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Okun B.

1 Scott R.

1 2011

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L²-homology and reciprocity for right-angled Coxeter groups