Locally compact lacunary hyperbolic groups (Q2013888)

Summary: 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 eld 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 nitely generated lacunary hyperbolic groups to the setting of locally compact groups. We moreover answer a question of \textit{A. Y. Ol'shanskii} et al. [Geom. Topol. 13, No. 4, 2051--2140 (2009; Zbl 1243.20056)] about subgroups of lacunary hyperbolic groups.