Lectures on Geometric Group Theory

Preface The main goal of this book is to describe several 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 one associates 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 somewhat broader viewpoint is to say that one studies a finitely generated group G by analyzing geometric properties of spaces X on which G acts geometrically, i.e., properly discontinuously, cocompactly and isometrically. The Cayley graph is just one of such spaces.) A primary obstacle 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 properties? The best one can hope here is to recover the group G up to virtual isomorphism from its geometry. Groups G 1 , G 2 are said to be virtually isomorphic if there exist subgroups Virtual isomorphism implies quasi-isometry but, in general, the converse is false, see Example 1.49.