Vanishing theorems and conjectures for the<i>ℓ</i><sup>2</sup>–homology of right-angled Coxeter groups

Davis, Michael W.; Okun, Boris

doi:10.2140/gt.2001.5.7

Cited by 73 publications

(122 citation statements)

References 31 publications

(78 reference statements)

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“…/. The sum is always finite because of (6). The conditions (a)-(d) are easy to check: (a) and (b) follow from (1), (3) and (6); (c) follows from (4) and (5): since the supports of H i .…”

Section: Chain Homotopy Contractionmentioning

confidence: 94%

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Thin buildings

Dymara

2006

Geom. Topol.

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Let X be a building of uniform thickness q C 1. L 2 -Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L 2 -cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q . The weighted cohomology makes sense for all real positive values of q , and is computed for small q . If the Davis complex of the Coxeter group is a manifold, a version of Poincaré duality allows to deduce that the L 2 -cohomology of a building with large thickness is concentrated in the top dimension.

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“…/. The sum is always finite because of (6). The conditions (a)-(d) are easy to check: (a) and (b) follow from (1), (3) and (6); (c) follows from (4) and (5): since the supports of H i .…”

Section: Chain Homotopy Contractionmentioning

confidence: 94%

“…We are interested in calculating these Betti numbers. (This problem was considered by Dymara and Januszkiewicz in [8] and by Davis and Okun in [6]. )…”

Section: Introductionmentioning

confidence: 99%

Thin buildings

Dymara

2006

Geom. Topol.

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“…Since this aspect is not widely known to combinatorialists, we make a brief digression to the corresponding problems. 50 …”

Section: De Nitions and Cubical Mapsmentioning

confidence: 99%

Torus Actions and Their Applications in Topology and Combinatorics

Buchstaber

Panov

2002

University Lecture Series

358

550

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Abstract. Here, the study of torus actions on topological spaces is presented as a bridge connecting combinatorial and convex geometry with commutative and homological algebra, algebraic geometry, and topology. This link helps in understanding the geometry and topology of a space with torus action by studying the combinatorics of the space of orbits. Conversely, the most subtle properties of a combinatorial object can be recovered by realizing it as the orbit structure for a proper manifold or complex acted on by a torus. The latter can be a symplectic manifold with Hamiltonian torus action, a toric variety or manifold, a subspace arrangement complement, etc., while the combinatorial objects include simplicial and cubical complexes, polytopes, and arrangements. This approach also provides a natural topological interpretation in terms of torus actions of many constructions from commutative and homological algebra used in combinatorics.The exposition centers around the theory of moment-angle complexes, providing an e ective way to study triangulations by methods of equivariant topology. The book includes many new and well-known open problems and would be suitable as a textbook. We hope that it will be useful for specialists both in topology and in combinatorics and will help to establish even tighter connections between the subjects involved.

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“…In [21] the first and fourth authors studied this case when .W; S/ was right-angled. (Recall Definition 13.2: .W; S / is right-angled if m st D 2 or 1 for all pairs fs; tg of distinct elements in S .)…”

Section: Properties Of Weighted L 2 -Homology In the Right-angled Casementioning

confidence: 99%

“…(Recall Definition 13.2: .W; S / is right-angled if m st D 2 or 1 for all pairs fs; tg of distinct elements in S .) Much of [21] extends in a straightforward fashion from q D 1 to the case of a general q. The purpose of this section is to rewrite parts of [21] in the general case.…”

Section: Properties Of Weighted L 2 -Homology In the Right-angled Casementioning

confidence: 99%

WeightedL²–cohomology of Coxeter groups

et al. 2007

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Given a Coxeter system .W; S / and a positive real multiparameter q, we study the "weighted L 2 -cohomology groups," of a certain simplicial complex † associated to .W; S /. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to .W; S / and the multiparameter q. They have a "von Neumann dimension" with respect to the associated "Hecke-von Neumann algebra" N q . The dimension of the i -th cohomology group is denoted b For a certain range of q, we calculate these cohomology groups as modules over N q and obtain explicit formulas for the b i q . †/. The range of q for which our calculations are valid depends on the region of convergence of the growth series of W . Within this range, we also prove a Decomposition Theorem for N q , analogous to a theorem of L Solomon on the decomposition of the group algebra of a finite Coxeter group.

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Vanishing theorems and conjectures for theℓ²–homology of right-angled Coxeter groups

Cited by 73 publications

References 31 publications

Thin buildings

Thin buildings

Torus Actions and Their Applications in Topology and Combinatorics

WeightedL²–cohomology of Coxeter groups

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Vanishing theorems and conjectures for theℓ2–homology of right-angled Coxeter groups

Cited by 73 publications

References 31 publications

Thin buildings

Thin buildings

Torus Actions and Their Applications in Topology and Combinatorics

WeightedL2–cohomology of Coxeter groups

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Product

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Vanishing theorems and conjectures for theℓ²–homology of right-angled Coxeter groups

WeightedL²–cohomology of Coxeter groups