Boris Okun

Vanishing theorems and conjectures for the ℓ2–homology of right-angled Coxeter groups

Associated to any nite flag complex L there is a right-angled Coxeter group WL and a cubical complex L on which WL acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of L is L and (2) L is contractible. It follows that if L is a triangulation of S n 1 ,t hen L is a contractible n{manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced ‘ 2 {homology except in the middle dimension) in the case of L where L is a triangulation of S n 1 . The program succeeds when n 4. This implies the Charney{Davis Conjecture on flag triangulations ofS 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.

Cohomology computations for Artin groups, Bestvina–Brady groups, and graph products

We compute: * the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), Bestvina-Brady groups, and graph products of groups, * the L^2-Betti numbers of Bestvina-Brady groups and of graph products of groups, * the weighted L^2-Betti numbers of graph products of Coxeter groups. In the case of arbitrary graph products there is an additional proviso: either all factors are infinite or all are finite.(However, for graph products of Coxeter groups this proviso is unnecessary.)

Weighted

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” Nq. The dimension of the i‐th cohomology group is denoted b i .†/. It is a nonnegative real number which varies continuously with q. When q is integral, the b i .†/ are the usual L 2 ‐Betti numbers of buildings of type .W;S/ and thickness q. For a certain range of q, we calculate these cohomology groups as modules over Nq and obtain explicit formulas for the b i .†/. 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 Nq, analogous to a theorem of LSolomon on the decomposition of the group algebra of a finite Coxeter group.

L^2

We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.

–cohomology of Coxeter groups

For any Coxeter group W, we define a filtration of H^*(W;ZW) by W-submodules and then compute the associated graded terms. More generally, if U is a CW complex on which W acts as a reflection group we compute the associated graded terms for H_*(U) and, in the case where the action is proper and cocompact, for H^*_c(U).

The strong Atiyah conjecture for right-angled Artin and Coxeter groups

We compute the compactly supported cohomology of the standard realization of any locally finite building. MathematicsSubjectClassification(2010). Primary 20F65; Secondary 20E42, 20F55, 20J06, 57M07.

Cohomology of Coxeter groups with group ring coefficients: II

We construct a tangential map from a locally symmetric space of noncompact type to its dual compact type twin. By comparing the induced map in cohomology to a map defined by Matsushima, we conclude that in the equal rank case the map has a nonzero degree. AMS Classification 53C35 ; 57T15, 55R37, 57R99

Compactly supported cohomology of buildings

The action dimension of a discrete group is the smallest dimension of a contractible manifold which admits a proper action of . Associated to any flag complex L there is a right-angled Artin group, AL. We compute the action dimension of AL for many L. Our calculations come close to confirming the conjecture that if an l 2 -Betti number of AL in degree l is nonzero, then the action dimension of AL is ≥ 2l. AMS classification numbers. Primary: 57Q15, 57Q25, 20F65, Secondary: 57R58

Nonzero degree tangential maps between dual symmetric spaces

We study group actions on manifolds that admit hierarchies, which generalizes the idea of Haken n-manifolds introduced by Foozwell and Rubinstein. We show that these manifolds satisfy the Singer conjecture in dimensions

The action dimension of right-angled Artin groups

n \le 4