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 G and a collection of subgroups S \mathcal S we say that S \mathcal S satisfies the limit set intersection property if for all H , K ∈ S H,K \in \mathcal S we have Λ ( H ) ∩ Λ ( K ) = Λ ( H ∩ K ) \Lambda (H)\cap \Lambda (K)=\Lambda (H\cap 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.