WeightedL²–cohomology of Coxeter groups

Abstract

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.

Factoriality of Hecke–von Neumann algebras of right-angled Coxeter groups

“…A generalization of the Singer Conjecture [4,Conjecture 14.7] deals with vanishing of certain weighted cohomology spaces of W. The motivation for this work was to try to approach this problem using the central decomposition of N q (W) to better understand the structure of these spaces. As it turns out, although the centers of N q (W) can be nontrivial, they contribute nothing new in the subject of decomposing the weighted cohomology of W; the decompositions described in [4,Theorem 11.1] are finer than those induced by the central decomposition of N q (W).…”

“…As it turns out, although the centers of N q (W) can be nontrivial, they contribute nothing new in the subject of decomposing the weighted cohomology of W; the decompositions described in [4,Theorem 11.1] are finer than those induced by the central decomposition of N q (W).…”

The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains

“…In works by M. Davis et al [18,17] the whole Poincaré series, not only its value at a point, is interpreted in terms of the weighted cohomology of Coxeter groups. The initial goal of this note was to give an e x p l i c i t expression not only of the zeros of these rational functions (and try to compare them with the eigenvalues of the Coxeter transformations) but also of their poles (not spoken about in [53,54,4] at all) in the particular cases of the (quasi-)Lannér groups, i.e., Coxeter groups (G, S) with (quasi-)Lannér Coxeter diagrams.…”

“…For applications of Poincaré series of the Coxeter groups of spherical or Euclidean type in the theory of simple finite groups, see [53]. There are other types of applications of the Poincaré series of the hyperbolic groups, see, e.g., [2,26,18].…”

“…Observe that in the above definition certain restrictions are taken for granted: the dimension of each homogeneous component must be finite, and only non-negative components are usually nonzero; "graded" is only assumed to be by means of Z; for Z k -graded objects (under similar restrictions: The support of the degrees with non-empty components lies in the cone with non-negative coordinates and each component is finite), we get series in several indeterminates, as in [42,18] and Table 2.…”