Elementary Subgroups of Relatively Hyperbolic Groups and Bounded Generation

Let G be a group hyperbolic relative to a collection of subgroups H λ, λ ∈ Λ. We say that a subgroup Q ≤ G is hyperbolically embedded into G, if G is hyperbolic relative to H λ, λ ∈ Λ ∪ Q. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element g ∈ G has infinite order and is not conjugate to an element of some H λ, λ ∈ Λ, then the (unique) maximal elementary subgroup containing g is hyperbolically embedded into G. This allows us to prove that if G is boundedly generated, then G is elementary or H λ = G for some λ ∈ Λ.