The L^2-Cohomology of Discrete Groups

Abstract

Given a space with a proper, cocompact group action, the L^2-cohomology groups are a particularly interesting invariant that incorporates the topology of the space and the geometry of the group action. We are interested in both the algebraic and geometric aspects of these invariants. From the algebraic side, the Strong Atiyah Conjecture claims that the L^2-Betti numbers assume only rational values, with certain prescribed denominators related to the torsion subgroups of the group. We prove this conjecture for the class of virtually cocompact special groups. This implies the Zero Divisor Conjecture holds for such groups. On the geometric side, the Action Dimension Conjecture claims that a group with that acts properly on a contractible n-manifold has vanishing L^2-Betti numbers above the middle dimension. We will prove this conjecture for many classes of right-angled Artin groups and Coxeter groups