Splittings and automorphisms of relatively hyperbolic groups

Abstract

We study automorphisms of a relatively hyperbolic group G. When G is oneended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL n (Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GL n (Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups.Given a malnormal quasiconvex subgroup P of a hyperbolic group G, we view G as hyperbolic relative to P and we apply the previous analysis to describe the group Out(P G) of automorphisms of P that extend to G: it is virtually a McCool group. If Out(P G) is infinite, then P is a vertex group in a splitting of G. If P is torsion-free, then Out(P G) is of type VF, in particular finitely presented.We also determine when Out(G) is infinite, for G relatively hyperbolic. The interesting case is when G is infinitely-ended and has torsion. When G is hyperbolic, we show that Out(G) is infinite if and only if G splits over a maximal virtually cyclic subgroup with infinite center. In general we show that infiniteness of Out(G) comes from the existence of a splitting with infinitely many twists, or having a vertex group that is maximal parabolic with infinitely many automorphisms acting trivially on incident edge groups.

McCool groups of toral relatively hyperbolic groups

“…A trivial but important remark is that T ⊂ Out(G; H (t) , K (t) ). As pointed out in Lemma 2.10 of [GL12], we have Given an edge e of Γ, there is a natural map ρ e : Out 0 (T ) → Out(G e ), defined in the same way as ρ v above. If v is an endpoint of e, the inclusion of G e into G v induces a homomorphism ρ v,e : Out(G v ; Inc v ) → Out(G e ) with ρ e = ρ v,e • ρ v (it is well-defined because the normalizer N Gv (G e ) acts on G e by inner automorphisms).…”

“…If v is a rigid vertex, then G v does not split over an abelian group relative to Inc v ∪ H ||Gv . By the Bestvina-Paulin method and Rips theory, one deduces that the image of Out 0 (T ) ∩ Out(G; H (t) ) in Out(G v ) is finite if H is a finite family of finitely generated subgroups (see [GL12], Theorem 3.9 and Proposition 4.7).…”

“…Edge groups being abelian, hence relatively quasiconvex, every vertex group G v is toral relatively hyperbolic (see for instance [GL12]). …”

“…||Gv ) with finite index (see [GL12], Proposition 4.7). If v is a rigid vertex, then G v does not split over an abelian group relative to Inc v ∪ H ||Gv .…”

Vertex finiteness for splittings of relatively hyperbolic groups

“…In a relatively hyperbolic group, all finite subgroups outside of a finite number of conjugacy classes are parabolic (see for instance Lemma 3.1 of [15]), so Assertion 1 follows from Lemma 2.1. Assertion 2 follows from Lemma 2.2.…”