Abstract

This book is based upon a set of lecture notes for a course that I was teaching at the University of Utah in Fall of 2002. Our main goal is to describe various tools of the quasi-isometric rigidity and to illustrate them by presenting (essentially self-contained) proofs of several fundamental theorems in this area: Gromov’s theorem on groups of polynomial growth, Mostow Rigidity Theorem and Schwartz’s quasi-isometric rigidity theorem for nonuniform lattices in the real-hyperbolic spaces. We conclude with a survey of the quasi-isometric rigidity theory. The main idea of the geometric group theory is to treat finitely-generated groups as geometric objects: with each finitely-generated group G we associate a metric space, the Cayley graph of G. One of the main issues of the geometric group theory is to recover as much as possible algebraic information about G from the geometry of the Cayley graph. A primary obsticle for this is the fact that the Cayley graph depends not only on G but on a particular choice of a generating set of G. Cayley graphs associated with different generating sets are not isometric but quasi-isometric. The fundamental question which we will try to address in this book is: If G,G ′ are quasi-isometric groups, to which extent G and G ′ share the same algebraic properies?

Citations