l_2-homology of right-angled Coxeter groups

Vanderbilt University, Nashville TN

Investigators

Abstract

Proposal: DMS-0072639 PI: Boris Okun Abstract Okun proposes to calculate l_2-homology of right-angled Coxeter groups. In a recent joint work (still in preparation) with M. Davis (the Ohio State University) we calculated l_2-homology of some right-angled Coxeter groups and, in particular, proved Singer's vanishing conjecture for right-angled Coxeter group manifolds of dimension <5. As a corollary we get that Hopf's conjecture about the sign of Euler characteristic of aspherical manifolds is true for non-positively curved (in the sense of Aleksandrov) 4-dimensional cubical manifolds. Our calculations in higher dimensions are consistent with Singer's vanishing conjecture. Several conjectures, related to Singer's vanishing conjecture are discussed. Coxeter groups, also known as reflection groups, play an important role in various branches of mathematics. The simplest examples of such groups come from Escher type tilings - tilings of the hyperbolic plane by convex polygons. The higher dimensional analogs of these give a large class of examples in topology and geometry of manifolds. The proposed research concentrates on several longstanding conjectures, as applied to these examples. A surprising corollary of this research is an explicit estimate for the genus of a graph - the minimal genus of the surface in which the graph can be embedded.