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 o...

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces U, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry ISO(U) contains everywhere dense locally finite subgroup; Theorem G(Globalization): For each finite metric space F there exists another finite metric space F̄ and isometric imbedding j of F to ...

2 Rigidity of non-uniform rank one lattices 6 2.1 Theorems of Richard Schwartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Finite volume real hyperbolic manifolds . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . ...