Box spaces: bridging geometric and asymptotic group theory

Abstract
If G is a finitely generated, residually finite group and $(N_i)$ is a sequence of finite index normal subgroups decreasing to the identity, the associated box spaces is the disjoint union of the $G/N_i$'s, with a suitable metric. Any box space of a property (T) group is an expander, by a result of Margulis (1973). An important question in geometric group theory is: which group properties are captured by a box space, up to coarse equivalence? The talk will survey the known results, ending with work in progress with Ana Khukhro.