Quasi-isometric rigidity in low dimensional topology

The early work of Mostow, Margulis and Prasad on rigidity of arithmetic lattices has evolved into a broad use of quasi-isometry techniques in group theory and low dimensional topology. The word metric on a finitely generated group makes it into a metric space which is uniquely determined up to the geometric relation called quasi-isometry, despite the fact that the metric depends on the choice of generating set. As for lattices in suitable Lie groups, where quasi-isometry of lattices implies commensurability, the general quasi-isometric study of groups aims to understand the remarkable extent to which this completely geometric notion often captures algebraic properties of the group. The Milnor-Schwarz Lemma provides an equivalence between the geometry of the word metric on the fundamental group of a compact Riemannian manifold (or metric complex) with the geometry of its universal cover. So the quasi-isometry study of groups also returns information about the spaces. This relationship has proved particularly productive in low dimensional geometry/topology.