Vanishing theorems and conjectures for the, $\ell ^2$--homology of right-angled Coxeter groups
By Michael W. Davis, B. Okun. 2001
115 citations recorded by the source corpus.
Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a cubical complex \Sigma_L on which W_L acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of \Sigma_L is L and (2) \Sigma_L is contractible. It follows that if L is a triangulation of S^{n-1}, then \Sigma_L is a contractible n-manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced L^2-homology except in the middle dimension) in the case of \Sigma_L where L is a triangulation of S^{n-1}. The program succeeds when n < 5. This implies the Charney-Davis Conjecture on flag triangulations of S^3. It also implies the following special case of the Hopf-Chern Conjecture: every closed 4-manifold with a nonpositively curved, piecewise Euclidean, cubical structure has nonnegative Euler characteristic. Our methods suggest the following generalization of the Singer Conjecture. Conjecture: If a discrete group G acts properly on a contractible n-manifold, then its L^2-Betti numbers b_i^{(2)} (G)$ vanish for i>n/2.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9413eae5520c5b87d5a509f82e2c40ea42202526
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Citations in the corpus
- An introduction to right-angled Artin groups (2006, unknown): “For progress on these and related conjectures, see [42], [18], and [74].”
- The Geometry and Topology of Coxeter Groups (2025, methodology): “In [91] it is suggested that there is a close analogy between Coxeter groups of type HM, with n odd, and 3-manifold theory (at least in the right-angled case).”
- Torus Actions and their Applications in Topology and Combinatorics (2002, background): “Is it true that the Euler characteristic (M2n) satis es the inequality ( 1)q (M2q) > 0? More details about the connection between the Charney{Davis and Hopf conjectures can be found in [41] and in a more recent paper [50].”
- Cohomology computations for Artin groups, Bestvina-Brady groups, and graph products (2010, background): “Some partial results and conjectures can be found in [19].”