Realising quasi-isometry groups

The quasi-isometry group QI(X) of a metric space X is a natural group of automorphisms of the space that preserve its large-scale structure. The quasi-isometry groups of most familiar spaces are usually enormous and quite wild. Spaces X for which QI(X) is understood tend to exhibit a sort of rigidity phenomenon: every quasi-isometry of such spaces is close to an isometry. We exploit this phenomenon to address the question of which abstract groups arise as the quasi-isometry groups of metric spaces. This talk is based on joint work with Paula Heim and Joe MacManus.