Automorphisms of shift spaces and the Higman - Thompson groups : the one-sided case

Funding: The authors are all grateful for support from EPSRC research grant EP/R032866/1; the third author also gratefully acknowledges support from Leverhulme Trust Research Project Grant RPG-2017-159.Let 1 ≤ r < n be integers. We give a proof that the group Aut(Xℕn,σn) of automorphisms of the one-sided shift on n letters embeds naturally as a subgroup ℋn of the outer automorphism group Out(Gn,r) of the Higman-Thompson group Gn,r. From this, we can represent the elements of Aut(Xℕn,σn) by finite state non-initial transducers admitting a very strong synchronizing condition. Let H ∈ ℋn and write |H| for the number of states of the minimal transducer representing H. We show that H can be written as a product of at most |H| torsion elements. This result strengthens a similar result of Boyle, Franks and Kitchens, where the decomposition involves more complex torsion elements and also does not support practical a priori estimates of the length of the resulting product. We also explore the number of foldings of de Bruijn graphs and give acounting result for these for word length 2 and alphabet size n. Finally, we offer new proofs of some known results about Aut(Xℕn,σn).Publisher PDFPeer reviewe

Bi-Amalgamated algebras along ideals

Let $f: A\rightarrow B$ and $g: A\rightarrow C$ be two commutative ring homomorphisms and let $J$ and $J'$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}(J)=g^{-1}(J')$. The \emph{bi-amalgamation} of $A$ with $(B, C)$ along $(J, J')$ with respect to $(f,g)$ is the subring of $B\times C$ given by $$A\bowtie^{f,g}(J,J'):=\big\{(f(a)+j,g(a)+j') \mid a\in A, (j,j')\in J\times J'\big\}.$$ This paper investigates ring-theoretic properties of \emph{bi-amalgamations} and capitalizes on previous works carried on various settings of pullbacks and amalgamations. In the second and third sections, we provide examples of bi-amalgamations and show how these constructions arise as pullbacks. The fourth section investigates the transfer of some basic ring theoretic properties to bi-amalgamations and the fifth section is devoted to the prime ideal structure of these constructions. All new results agree with recent studies in the literature on D'Anna-Finocchiaro-Fontana's amalgamations and duplications.Comment: 15 page