2 00 4 Elementary subgroups of relatively hyperbolic groups and bounded generation

Elementary subgroups of relatively hyperbolic groups and bounded generation. Abstract 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 H λ , λ ∈ Λ, then the (unique) maximal elementary subgroup contained g is hyperbolically embedded into G. This allows to prove that if G is boundedly generated, then G is elementary or H λ = G for some λ ∈ Λ.