Low degree bounded cohomology and l^2-invariants for negatively curved groups

Abstract

We study the subgroup structure of discrete groups which share cohomological properties which resemble non-negative curvature. Examples include all Gromov hyperbolic groups. We provide strong restrictions on the possible s-normal subgroups of a Gromov hyperbolic group, or more generally a 'negatively curved' group. Another result says that the image of a group, which is boundedly generated by a finite set of amenable subgroups, in a Gromov hyperbolic group has to be virtually cyclic. Moreover, we show that any homomorphic image of an analogue of a higher rank lattices in a Gromov hyperbolic group must be finite. These results extend to a certain class of randomorphisms in the sense of Monod. We study the class of groups which admit proper quasi-1-cocycles and show that it is closed under l2-orbit equivalence.