Abstract

Given a metric space X, we denote by QI(X) the set of all quasi-isometries f : X → X, modulo finite sup-distance. This set admits a natural group structure via composition, and is called the full quasi-isometry group of X. These groups are, in general, incredibly wild and hard to compute, even for very natural spaces, and very few explicit examples are known. One source of explicit examples comes from certain families of symmetric spaces, due to a strong rigidity theorem of Pansu.  In this talk I will discuss how, given any countable group G, one can apply Pansu’s rigidity theorem together with the classical Frucht’s theorem from graph theory, and construct uncountably many quasi-isometry classes of metric spaces X such that QI(X) = G. I will also advertise some interesting open problems related to QI groups. This talk is based on joint work with Paula Heim and Lawk Mineh.