Automorphism groups of tree actions and of graphs of groups

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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 ο 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 Θ. We produce an exact sequence 1 → In Aut(Θ) → Aut(Θ) → 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 Θ. In the process we obtain similar information about the structure of Aut(Θ). We also draw the consequences in the case of amalgams and HNN-extensions.

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Journal of Pure and Applied Algebra