Graphs of hyperbolic groups and a limit set intersection theorem

Abstract

We define the notion of limit set intersection property for a collection of subgroups of a hyperbolic group; namely, for a hyperbolic group G and a collection of subgroups S we say that S satisfies the limit set intersection property if for all H,K ε S we have Λ(H)n∩(K) = Λ(Hn∩K). Given a hyperbolic group admitting a decomposition into a finite graph of hyperbolic groups structure with QI embedded condition, we show that the set of conjugates of all the vertex and edge groups satisfies the limit set intersection property.