Random groups of intermediate rank

I will discuss a new construction of nonpositively curved spaces whose rank is "intermediate" between spaces of rank 1 (like trees, hyperbolic spaces,...) and spaces of higher rank (such as the symmetric space of $SL_n$, $n>2$). Associated to these spaces are finitely presented groups which are almost, but not quite, uniform lattices in certain algebraic groups, typically $SL_3(K)$, $K$ a local field. The constructions use techniques from Gromov's well-known models of random groups. The first part of the talk will be devoted to explaining the idea of rank interpolation in general.