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.

Splittings and automorphisms of relatively hyperbolic groups

“…Conversely, one may add to P a finite subgroup or a maximal loxodromic subgroup (see e.g. [Osi06a]). These operations do not change the set of elementary (or relatively quasiconvex, as defined below) subgroups, and it is sometimes convenient (as in [Hru10]) to assume that every P i is infinite.…”

“…If G is (absolutely) hyperbolic, and P is a subgroup, then G is hyperbolic relative to {P } if (and only if) P is quasiconvex and almost malnormal, see [Bow12,Theorem 7.11] or [Osi06a]. If so, Theorem 6.2 applies and describes Out(P G), the automorphisms of P which extend to G. Corollary 6.3.…”

Peripheral fillings of relatively hyperbolic groups

“…We say that a subgroup H of a group G almost has CEP if there is a finite set of non-trivial elements F ⊆ H such that H ∩ N G = N whenever N ∩ F = ∅. Recall that a subgroup H of a group G is said to be almost malnormal, if H g ∩ H is finite for all g / ∈ H. Bowditch [2] proved that if G is a hyperbolic group and H is an almost malnormal quasi-convex subgroup of G, then G is hyperbolic relative to H (see also [18]). Thus the following is an immediate corollary of Theorem 1.1.…”

Malnormal subgroups and Frobenius groups: basics and examples

“…A group is elementary if it contains a cyclic subgroup of finite index. It is proved in [32] that, for every loxodromic element g ∈ G, there is a unique maximal elementary subgroup E G (g) G containing g. Two loxodromic elements f, g are commensurable (in G) if f k is conjugate to g l in G for some non-zero k, l. A subgroup S of G is called suitable if it contains two non-commensurable loxodromic elements g, h such that E G (g) ∩ E G (h) = {1}.…”