  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.
