arXiv subjects

Feyishayo Olukoya

Publications and source records attributed to Feyishayo Olukoya.

11 recordsLinked to original sources

arXiv2024

Conjugacy for certain automorphisms of the one-sided shift via transducers

We address the following open problem, implicit in the 1990 article "Automorphisms of one-sided subshifts of finite type" of Boyle, Franks and Kitchens (BFK): "Does there exists an element $ψ$ in the group of automorphisms of the one-sided shift $\operatorname{Aut}(\{0,1,\ldots,n-1\}^{\mathbb{N}}, σ_{n})$ so that all points of $\{0,1,\ldots,n-1\}^{\mathbb{N}}$ have orbits of length $n$ under $ψ$ and $ψ$ is not conjugate to a permutation?" Here, by a 'permutation' we mean an automorphism of one-sided shift dynamical system induced by a permutation of the symbol set $\{0,1,\ldots,n-1\}$. We resolve this question by showing that any $ψ$ with properties as above must be conjugate to a permutation. Our techniques naturally extend those of BFK using the strongly synchronizing automata technology developed here and in several articles of the authors and collaborators (although, this article has been written to be largely self-contained).

math.GR↗

arXiv2024

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 $\mathcal{L}_{n,r}$ of the outer-automorphism groups $\mathcal{O}_{n,r}$ of the Higman--Thompson group $G_{n,r}$, and to extend results and techniques in $\operatorname{Aut}(X_n^{\mathbb{Z}}, σ_{n})$ to the groups of automorphisms $\operatorname{Aut}(G_{n,r})$ and outerautomrphisms of the Higman--Thompson group $G_{n,r}$. 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 $\mathcal{K}_{n}$ of $\mathcal{L}_{n,n-1}$. Using this realisation, we show that the $\operatorname{Aut}(G_{n,r})$ contains an isomorphic copy of $\operatorname{Aut}(X_{m}^{\mathbb{Z}}, σ_{m})$ for all $m \ge 2$. A survey of the literature then yields that $\operatorname{Aut}(G_{n,r})$ 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 $\operatorname{Aut}(X_n^{\mathbb{Z}}, σ_{n})$ to the group $\mathcal{O}_{n,n-1}$. The homeomorphism $\overleftarrow{\phantom{a}}$ of $X_n^{\mathbb{Z}}$ which maps a sequence $(x_i)_{i \in \mathbb{Z}}$ to the sequence $(y_{i})_{i \in \mathbb{Z}}$ defined such that $y_{i} = x_{-i}$ induces an automorphism $\overleftarrow{\mathfrak{r}}$ of $\operatorname{Aut}(X_n^{\mathbb{Z}}, σ_{n})$, and consequently, an automorphism of $\mathcal{L}_{n}$. We extend the automorphism $\overleftarrow{\mathfrak{r}}$ to the group $\mathcal{O}_{n,n-1}$. In a forthcoming article, we demonstrate that the group $\mathcal{O}_{n}$ is isomorphic to the mapping class group of the full two-sided shift over $n$ letters.

math.GR↗

arXiv2021

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

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 $\{G_{n,r}\}$. We show that the quotient of the group of automorphisms of the (two-sided) shift dynamical system $\mathrm{Aut}(X_n^{\mathbb{N}}, σ_{n})$ by its centre embeds as a particular subgroup $\mathcal{L}_{n}$ of the outer automorphism group $\mathop{\mathrm{Out}}(G_{n,n-1})$ of $G_{n,n-1}$. It follows by a result of Ryan that we have the following central extension: $$\langle σ_{n}\rangle \hookrightarrow \mathrm{Aut}(X_n^{\mathbb{N}}, σ_{n}) \twoheadrightarrow \mathcal{L}_{n}$$ where here, $\langle σ_{n} \rangle \cong \mathbb{Z}$. 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 $1 \le r < n$, the groups $\mathop{\mathrm{Out}}(G_{n,r})$ are centreless and have undecidable order problem. Note that the group $\mathop{\mathrm{Out}}(G_{n,n-1})$ consists of finite transducers (combinatorial objects arising in automata theory), and elements of the group $\mathcal{L}_{n}$ are easily characterised within $\mathop{\mathrm{Out}}(G_{n,n-1})$ 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 $\mathrm{Aut}(X_n^{\mathbb{N}}, σ_{n})$, and we demonstrate how to compute products using this new representation.

math.GR↗

arXiv2021

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

Let $1 \le r < n$ be integers. We give a proof that the group $\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, σ_{n}})$ of automorphisms of the one-sided shift on $n$ letters embeds naturally as a subgroup $\mathcal{H}_{n}$ of the outer automorphism group $\mathop{\mathrm{Out}}({G_{n,r}})$ of the Higman-Thompson group $G_{n,r}$. From this, we can represent the elements of $\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, σ_{n}})$ by finite state non-initial transducers admitting a very strong synchronizing condition. Let $H \in \mathcal{H}_{n}$ 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 \textit{a priori} estimates of the length of the resulting product. We also explore the number of foldings of de Bruijn graphs and give a counting result for these for word length $2$ and alphabet size $n$. Finally, we offer new proofs of some known results about $\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, σ_{n}})$.

math.GR↗

arXiv2020

Automorphism towers of groups of homeomorphisms of Cantor space

We show that for any full and sufficiently transitive (i.e. \textit{flexible}) group $G$ of homeomorphisms of Cantor space, $\mathrm{Aut}(\mathrm{Aut}(G)) = \mathrm{Aut}(G)$. This class contains many generalisations of the Higman-Thompson groups $G_{n,r}$, and the Rational group $\mathcal{R}_{2}$ of Grigorchuk, Nekrashevych, and Suchanski{\u ı}. We also demonstrate that for generalisations $T_{n,r}$ of R. Thompson's group $T$, $\mathrm{Aut}(\mathrm{Aut}(T_{n,r}))= \mathrm{Aut}(T_{n,r})$. In the case of the groups $G_{n,r}$ and $T_{n,r}$ our results extend results of Brin and Guzm{\' a}n for Thompson's group $T$, and generalisations of Thompson's group $F$.

math.GR↗

arXiv2020

The growth rates of automaton groups generated by reset automata

We give sufficient conditions for when groups generated by automata in a class $\mathcal{C}$ of transducers, which contains the class of reset automata transducers, have infinite order. As a consequence we also demonstrate that if a group generated by an automata in $\mathcal{C}$ is infinite, then it contains a free semigroup of rank at least 2. This gives a new proof, in the context of groups generated by automaton in $\mathcal{C}$, of a result of Chou showing that finitely generated elementary amenable groups either have polynomial growth or contain a free semigroup of rank at least 2.

math.GR↗

arXiv2020

An automata theoretic proof that $\mathop{\mathrm{Out}}(T) \cong \mathbb{Z}/2\mathbb{Z}$ and some embedding results for $\mathop{\mathrm{Out}}(V)$

In a seminal paper, Brin demonstrates that the outerautomorphism group of Thompson group $T$ is isomorphic to the cyclic group of order two. In this article, building on characterisation of automorphisms of the Higman-Thompson groups $G_{n,r}$ and $T_{n,r}$ as groups of transducers, we give a new proof, automata theoretic in nature, of Brin's result. We also demonstrate that the group of outerautomorphisms of Thompson's group $V = G_{2,1}$ contains an isomorphic copy of Thompson's group $F$. This extends a result of the author demonstrating that whenever $n \ge 3$ and $1 \le r < n$ the outerautomorphism groups of $G_{n,r}$ and $T_{n,r}$ contain an isomorphic copy of $F$.

math.GR↗

arXiv2020

The core growth of strongly synchronizing transducers

We introduce the notion of `core growth rate' for strongly synchronizing transducers. We explore some elementary properties of the core growth rate and give examples of transducers with exponential core growth rate. We conjecture that all strongly synchronizing transducers which generate an automaton group of infinite order have exponential core growth rate. There is a connection to the group of automorphisms of the one-sided shift. More specifically, the results of this article are related to the question of whether or not there can exist infinite order automorphisms of the one-sided shift with infinitely many roots.

math.GR↗

arXiv2019

Automorphisms of the generalised Thompson's group $T_{n,r}$

The recent paper "The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups $G_{n,r}$" of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterises the automorphisms of the Higman-Thompson groups $G_{n,r}$ as the specific subgroup of the rational group $\mathcal{R}_{n,r}$ of Grigorchuk, Nekrashevych and Suchanski{\u i}'s consisting of those elements which have the additional property of being bi-synchronizing. In this article, we extend the arguments of BCMNO and characterise the automorphism group of $T_{n,r}$ as a subgroup of $\mathrm{Aut}{G_{n,r}}$. We then show that the groups $\mathrm{Out}{T_{n,r}}$ can be identified with subgroups of the group $\mathrm{Out}{T_{n,n-1}}$. Extending results of Brin and Guzman, we show that the groups $\mathrm{Out}{T_{n,r}}$, for $n>2$, are all infinite and contain an isomorphic copy of Thompson's group $F$. For $X \in \{T,G\}$, we study the groups $\mathrm{Out}{X_{n,r}}$ and show that these fit in a lattice structure where $\mathrm{Out}{X_{n,1}} \unlhd \mathrm{Out}{X_{n,r}}$ for all $1 \le r \le n-1$ and $\mathrm{Out}{X_{n,r}} \unlhd \mathrm{Out}{X_{n,n-1}}$. This gives a partial answer to a question in BCMNO concerning the normal subgroup structure of $\mathrm{Out}{G_{n,n-1}}$. Furthermore, we deduce that for $1\le j,d \le n-1$ such that $d = \gcd(j, n-1)$, $\mathrm{Out}{X_{n,j}} = \mathrm{Out}{X_{n,d}}$ extending a result of BCMNO for the groups $G_{n,r}$ to the groups $T_{n,r}$. We give a negative answer to the question in BCMNO which asks whether or not $\mathrm{Out}{G_{n,r}} \cong \mathrm{Out}{G_{n,s}}$ if and only if $\gcd(n-1,r) = \gcd(n-1,s)$. We conclude by showing that the groups $T_{n,r}$ have the $R_{\infty}$ property extending the result of Burillo, Matucci and Ventura and, independently, Gon{\c c}alves and Sankaran, for Thompson's group $T$.

math.GR↗

arXiv2019

The further chameleon groups of Richard Thompson and Graham Higman: Automorphisms via dynamics for the Higman groups $G_{n,r}$

We describe, through the use of Rubin's theorem, the automorphism groups of the Higman-Thompson groups $G_{n,r}$ as groups of specific homeomorphisms of Cantor spaces $\mathfrak{C}_{n,r}$. This continues a thread of research begun by Brin, and extended later by Brin and Guzmán: to characterise the automorphism groups of the `Chameleon groups of Richard Thompson,' as Brin referred to them in 1996. The work here completes the first stage of that twenty-year-old program, containing (amongst other things) a characterisation of the automorphism group of $V$, which was the `last chameleon.' The homeomorphisms which arise fit naturally into the framework of Grigorchuk, Nekrashevich, and Suschanskii's rational group $\mathscr{R}$: they are exactly those homeomorphisms which are induced by bi-sychronizing transducers, which we define in the paper. This result appears to offer insight into the nature of Brin and Guzman's exotic automorphisms, while also uncovering connections with the theory of reset words for automata (arising in the Road Colouring Problem) and with the theory of automorphism groups of the full shift.

math.GR↗

arXiv2018

Conjugate subgroups and overgroups of $V_n$

We describe subgroups and overgroups of the generalised Thompson groups $V_n$ which arise via conjugation by rational homeomorphisms of Cantor space. We specifically consider conjugating $V_n$ by homeomorphisms induced by synchronizing transducers and their inverses. Our descriptions of the subgroups and overgroups use properties of the conjugating transducer to either restrict or augment the action of $V_n$ on Cantor space.

math.GR↗