Alexander Olshanskii

Abstract
The talk will be based on recent results obtained with Denis Osin and Mark Sapir. We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an $\R$-tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants.Using lacunary hyperbolic groups we solve a number of problems on asymptotic cones of finitely generated groups.