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# Lacunary hyperbolic groups

@inproceedings{Sapir2007LacunaryHG, title={Lacunary hyperbolic groups}, author={Mark V. Sapir}, year={2007} }

- Published 2007

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 and injectivity radii. Using central extensions of lacunary hyperbolic groups, we solve a problem of Gromov by constructing a group whose asymptotic cone C has countable but non-trivial fundamental group (in fact C is homeomorphic to the direct… CONTINUE READING