Automorphism groups of tree actions and of graphs of groups

Abstract

Let Γ be a group. The minimal non-abelian Γ-actions on real trees can be parametrized by the projective space of the associated length functions. The outer automorphism group of Γ, Out(Γ) = Aut(Γ) ad(Γ) , acts on this space. Our objective is to calculate the stabilizer Out( Γ) l = { α ∈ Aut( Γ) > l ∘ α = l}/ad( Γ), where l is the length function of a minimal non-abelian action (without inversion) on a simplicial tree. In this case, stabilizing l up to a scalar factor is equivalent to stabilizing l. The simplicial tree action is encoded by a quotient graph of groups U . We produce an exact sequence 1 → In Aut( U ) → Aut( U ) → Out( Γ) l → 1. A six-step filtration on Out( Γ) l is obtained, where successive quotients are explicitly described in terms of the data defining U . In the process we obtain similar information about the structure of Aut( U ). We also draw the consequences in the case of amalgams and HNN-extensions.