Separating quasi-convex subgroups in 7-systolic groups

Abstract

Let Γ be a group acting without inversions and simply transitively on the top-dimensional simplices of some simply-connected simplicial complex X with “simplicial negative curvature”. Then the quasi-convex subgroups of Γ are convex-cocompact. Furthermore, if the action of Γ on X satisfies some additional condition called “extra-tilability”, the quasi-convex subgroups of Γ are separable, i.e., every such subgroup is the intersection of finite index subgroups. The latter result applies to a large class of “simplicially negatively curved” groups recently constructed by Januszkiewicz and the second author.