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 L₍, ₑ of the outer-automorphism groups O₍, ₑ of the Higman--Thompson group G₍, ₑ, and to extend results and techniques in Aut (Xₙ^Z, ₍) to the groups of automorphisms Aut (G₍, ₑ) and outerautomrphisms of the Higman--Thompson group G₍, ₑ. 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 K₍ of L₍, ₍-₁. Using this realisation, we show that the Aut (G₍, ₑ) contains an isomorphic copy of Aut (X₌^Z, ₌) for all m 2. A survey of the literature then yields that Aut (G₍, ₑ) 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 (Xₙ^Z, ₍) to the group O₍, ₍-₁. The homeomorphism a of Xₙ^Z which maps a sequence (xᵢ) ₈ ₙ to the sequence (y₈) ₈ ₙ defined such that y₈ = x-₈ induces an automorphism r of Aut (Xₙ^Z, ₍), and consequently, an automorphism of L₍. We extend the automorphism {r} to the group O₍, ₍-₁. In a forthcoming article, we demonstrate that the group O₍ is isomorphic to the mapping class group of the full two-sided shift over n letters.