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Locally compact lacunary hyperbolic groups

We investigate the class of locally compact lacunary hyperbolic groups. We prove that if a locally compact compactly generated group G admits one asymptotic cone that is a real tree and whose natural transitive isometric action is focal, then G must be a focal hyperbolic group. As an application, we characterize connected Lie groups and linear algebraic groups over an ultrametric local field of characteristic zero having cut-points in one asymptotic cone. We prove several results for locally compact lacunary hyperbolic groups, and extend the characterization of finitely generated lacunary hyperbolic groups to the setting of locally compact groups. We moreover answer a question of Olshanskii, Osin and Sapir about subgroups of lacunary hyperbolic groups.