When studying a metric space, it can be interesting to
consider the group of maps preserving its large scale geometry. These
maps are called quasiisometries and the associated group is called the
QI group. Determining the QI group of a metric space is, in general, a
hard problem. Few QI groups are known explicitly, and most of these
results arise from a phenomenon called QI rigidity, which essentially
says that QI(X)=Isom(X). In this talk we will explore these concepts and
give a partial answer to the question which groups can arise as QI
groups of metric spaces. This talk is based on joint work with Joe
MacManus and Lawk Mineh.