2020/06/02 by Belk, James, Bleak, Collin, Cameron, Peter J. +1 · 1 citation
#20E36 #20F10 #28D15 #54H15 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2006.01466
In this article, we further explore the nature of a connection between the groups of automorphisms of full shift spaces and the groups of outer automorphisms of the Higman--Thompson groups \Gn,r\. We show that the quotient of the group of automorphisms of the (two-sided) shift dynamical system Aut(Xnℕ, σn) by its centre embeds as a particular subgroup Ln of the outer automorphism group \mathopOut(Gn,n-1) of Gn,n-1. It follows by a result of Ryan that we have the following central extension: ⟨ σn⟩ \hookrightarrow Aut(Xnℕ, σn) \twoheadrightarrow Ln where here, ⟨ σn ⟩ ≅ ℤ. We prove that this short exact sequence splits if and only if n is not a proper power, and, in all cases, we compute the 2-cocycles and 2-coboundaries for the extension. We also use this central extension to prove that for 1 ≤ r < n, the groups \mathopOut(Gn,r) are centreless and have undecidable order problem. Note that the group \mathopOut(Gn,n-1) consists of finite transducers (combinatorial objects arising in automata theory), and elements of the group Ln are easily characterised within \mathopOut(Gn,n-1) by a simple combinatorial property. In particular, the short exact sequence allows us to determine a new and efficient purely combinatorial representation of elements of Aut(Xnℕ, σn), and we demonstrate how to compute products using this new representation.