We study the subgroup structure of discrete groups that 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 ’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 group,...

We study property RD in terms of rapid decay of matrix coefficients. We give new formulations of property RD in terms of a L-integrability condition of a Banach representation. Combining this new definition with the existence of cyclic subgroups of exponential growth in non-uniform lattices in semisimple Lie groups, we deduce that there exist matrix coefficients associated to several kinds of q...