Automorphisms of the two-sided shift and the Higman--Thompson groups III: extensions

We aim to interpret important constructions in the theory of automorphisms of the shift dynamical system in terms of subgroups Ln,r of the outer-automorphism groups On,r of the Higman--Thompson group Gn,r, and to extend results and techniques in Aut(XₙZ, σₙ) to the groups of automorphisms Aut(Gn,r) and outerautomrphisms of the Higman--Thompson group Gn,r. Our mains results are a concrete realisation of the "inert subgroup", important subgroup in the study of automorphism groups of shift spaces, as a subgroup Kₙ of Ln,n-1. Using this realisation, we show that the Aut(Gn,r) contains an isomorphic copy of Aut(XₘZ, σₘ) for all m ≥ 2. A survey of the literature then yields that Aut(Gn,r) contains isomorphic copies of finite groups, finitely generated abelian groups, free groups, free products of finite groups, fundamental groups of 2-manifolds, graph groups and countable locally finite residually finite groups to name a few. We extend a result for Aut(XₙZ, σₙ) to the group On,n-1. The homeomorphism a of XₙZ which maps a sequence (xᵢ)i ∈ Z to the sequence (yᵢ)i ∈ Z defined such that yᵢ = x₋ᵢ induces an automorphism r of Aut(XₙZ, σₙ), and consequently, an automorphism of Lₙ. We extend the automorphism r to the group On,n-1. In a forthcoming article, we demonstrate that the group Oₙ is isomorphic to the mapping class group of the full two-sided shift over n letters.