GRAPHS OF HYPERBOLIC GROUPS AND A LIMIT SET INTERSECTION THEOREM

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 is an element of S we have Lambda(H)boolean AND Lambda(K) = Lambda(H boolean AND 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.