In Gromov's density model for random groups at low densities groups can be cubulated (Ollivier-Wise) whereas at higher densities the groups have Property (T) (Zuk, Kotowski-Kotowski), but there's a gap where things are unclear.  I'll discuss two results, with Piotr Przytycki and Cornelia Drutu respectively, about this problem.

A surface group is the fundamental group of a closed surface of non-positive Euler characteristic.    A great deal of recent energy in geometric group theory has focussed on finding surface subgroups in various classes of groups of geometrical interest, especially word-hyperbolic groups. I will survey recent developments, highlights of which include Kahn—Markovic’s solution of the Surface Subgroup conjecture for Kleinian groups and Calegari—Walker’s discovery of surface subgroups in random groups