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 exist an element in the group of automorphisms of the one-sided shift Aut({0, 1, ... , n-1}N, sigma n) so that all points of {0, 1, ... , n-1}N have orbits of length n under and is not conjugate to a permutation? Here, by a permutation we mean an automorphism of the one-sided shift dynamical system induced by a permutation of the symbol set {0, 1, ... , n-1}. We resolve this question by showing constructively that any 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).

The 1973 Boone-Higman conjecture predicts that every finitely generated group with solvable word problem embeds in a finitely presented simple group. In this paper, we show that hyperbolic groups satisfy this conjecture, that is, each hyperbolic group embeds in some finitely presented simple group. This shows that the conjecture holds in the "generic" case for finitely presented groups. Our key tool is a new family of groups, which we call "rational similarity groups (RSGs)", that is interesting in its own right. We prove that every hyperbolic group embeds in a full, contracting RSG, and every full, contracting RSG embeds in a finitely presented simple group, thus establishing the result. Another consequence of our work is that all contracting self-similar groups satisfy the Boone-Higman conjecture.

The simple vigorous groups form a broad class of groups of homeomorphisms of Cantor space that includes Thompson's group V, its various generalisations and many others such as Nekrashevych's groups of dynamical origin. Bleak, Elliott and Hyde (2024) proved that every finitely generated simple vigorous group is 2-generated, and, in this paper, we give several strong generation results for this class of simple groups. For example, we prove that if G is a finitely generated simple vigorous group, then G is generated by three involutions, G is generated by an element of order m and an element of order n for any choice of m ≥ 2 and n ≥ 3, G has a minimal generating set of size k for all k ≥ 2, every nontrivial element of G is contained in a generating pair and the direct power G^n is 2-generated for all n. These results are analogous to well-known results for finite simple groups, but of course the proofs in this context are quite different. One consequence of our results is that Thompson's group V is (2, 3)-generated, which answers a question of Sapir (2017). Another consequence is that every finitely generated group quasi-isometrically embeds in a (2, 3)-generated simple group, strengthening theorems of Hall (1974) and Bridson (1998). All of our proofs are constructive, and we establish several new generation criteria for these groups, which we expect to be of wider interest, even just for Thompson's group V.

Every finite simple group can be generated by two elements and, in fact, every nontrivial element is contained in a generating pair. Groups with this property are said to be $${3 \over 2}$$ 3 2 -generated, and the finite $${3 \over 2}$$ 3 2 -generated groups were recently classified. Turning to infinite groups, in this paper, we prove that the finitely presented simple group T of Thompson is $${3 \over 2}$$ 3 2 -generated. Moreover, we exhibit an element ζ ∈ T such that for any nontrivial α ∈ T , there exists γ ∈ T such that 〈 α , ζ γ 〉 = T .

Every finite simple group can be generated by two elements and, in fact, every nontrivial element is contained in a generating pair. Groups with this property are said to be 32-generated, and the finite 32-generated groups were recently classified. Turning to infinite groups, in this paper, we prove that the finitely presented simple group T of Thompson is 32-generated. Moreover, we exhibit an element zeta is an element of T such that for any nontrivial alpha is an element of T, there exists gamma is an element of T such that = T.

We introduce the concept of a type system P \mathcal {P} , that is, a partition on the set of finite words over the alphabet { 0 , 1 } \{0,1\} compatible with the partial action of Thompson’s group V V , and associate a subgroup Stab V ⁡ ( P ) \operatorname {Stab}_{V}(\mathcal {P}) of V V . We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of V V . We also find an uncountable family of pairwise nonisomorphic maximal subgroups of V V . These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in V V of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of V V (both related to primitivity) are each satisfied only by V V itself, giving new ways to recognise when a subgroup of V V is not actually proper.

Every finite simple group can be generated by two elements and, in fact, every nontrivial element is contained in a generating pair. Groups with this property are said to be $\frac{3}{2}$-generated, and the finite $\frac{3}{2}$-generated groups were recently classified. Turning to infinite groups, in this paper, we prove that the finitely presented simple group $T$ of Thompson is $\frac{3}{2}$-generated. Moreover, we exhibit an element $\zeta \in T$ such that for any nontrivial $\alpha \in T$, there exists $\gamma \in T$ such that $\langle \alpha, \zeta^\gamma \rangle = T$.

A conjecture of Boone and Higman from the 1970's asserts that a finitely generated group G has solvable word problem if and only if G can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.

We show that R. Thompson’s group T is a maximal subgroup of the group V. The argument provides examples of foundational calculations which arise when expressing elements of V as products of transpositions of basic clopen sets in the Cantor space ℭ .

We introduce a new framework linking group theory and formal language theory which generalizes a number of ways these topics have been linked in the past. For a language class C in the Chomsky hierarchy, we say a group is epiC if it admits a language L over a finite (monoidal) generating set X ⊆ G in the class C such that the image of L under the evaluation map is G ∖{1_G}. We provide some examples of epiC groups and prove that the property of being epiC is not dependent on the generating set chosen. We also prove that epiC groups are closed under passage to finite index overgroups, taking extensions, and taking graph products of finitely many groups. Furthermore, we prove that epiRegular groups are closed under passage to finite index subgroups. Finally, we provide a characterization of the property of having solvable word problem within the framework of epiC groups.

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 Guzm\'an: 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.

Let $\mathfrak{C}$ be some Cantor space. We study groups of homeomorphisms of $\mathfrak{C}$ which are vigorous, or, which are flawless, where we introduce both of these terms here. We say a group $G\leq \operatorname{Homeo}(\mathfrak{C})$ is $vigorous$ if for any clopen set $A$ and proper clopen subsets $B$ and $C$ of $A$ there is $\gamma \in G$ in the pointwise-stabiliser of $\mathfrak{C}\backslash A$ with $B\gamma\subseteq C$. Being vigorous is similar in impact to some of the conditions proposed by Epstein in his proof that certain groups of homeomorphisms of spaces have simple commutator subgroups (and/or related conditions, as proposed in some of the work of Matui or of Ling). A non-trivial group $G\leq \operatorname{Homeo}(\mathfrak{C})$ is $flawless$ if for all $k$ and $w$ a non-trivial freely reduced product expression on $k$ variables (including inverse symbols), a particular subgroup $w(G)_\circ$ of the verbal subgroup $w(G)$ is the whole group. It is true, for instance, that flawless groups are both perfect and lawless. We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) vigorous groups are simple if and only if they are flawless, and, 3) the class of vigorous simple subgroups of $\operatorname{Homeo}(\mathfrak{C})$ is fairly broad (it contains many well known groups such as the commutator subgroups of the Higman-Thompson groups $G_{n,r}$, the Brin-Thompson groups $nV$, Rover's group $V(\Gamma)$, and others of Nekrashevych's `simple groups of dynamical origin', and, the class is closed under various natural constructions).

Let $m\leqslant n\in \mathbb {N}$, and $G\leqslant \operatorname {Sym}(m)$ and $H\leqslant \operatorname {Sym}(n)$. In this article, we find conditions enabling embeddings between the symmetric R. Thompson groups ${V_m(G)}$ and ${V_n(H)}$. When $n\equiv 1 \mod (m-1)$, and under some other technical conditions, we find an embedding of ${V_n(H)}$ into ${V_m(G)}$ via topological conjugation. With the same modular condition, we also generalize a purely algebraic construction of Birget from 2019 to find a group $H\leqslant \operatorname {Sym}(n)$ and an embedding of ${V_m(G)}$ into ${V_n(H)}$.

We were asked by Innovation Embassy to work with a large dataset centred around gambling investment, with the task of making a predictive function for computing Customer Lifetime Value (CLV), and also to see if there are ways of detecting fraudulent financial practices and addictive gambling patterns. We had moderate success with the data as it stands, but we were partly held back for two main reasons: the ability to discern a solid definition of CLV due to highly inconsistent data and data that contained many large and incomputable gaps. Different machine learning algorithms were used to find CLV functions based on key variables. We also describe a short and explicit list of ways where the base data can be improved to support effective calculation of CLV. Our key findings suggest that the average customer's CLV is 1035 and ~80% of revenue is brought in from ~10% of the clients.

The set of finitely generated subgroups of the group P L + ( I ) PL_+(I) of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R. Thompson’s group F F . Here, we show that every finitely generated subgroup G > P L + ( I ) G>PL_+(I) is either soluble, or contains an embedded copy of the finitely generated, non-soluble Brin-Navas group B B , affirming a conjecture of the first author from 2009. In the case that G G is soluble, we show the derived length of G G is bounded above by the number of breakpoints of any finite set of generators. We specify a set of ‘computable’ subgroups of P L + ( I ) PL_+(I) (which includes R. Thompson’s group F F ) and give an algorithm which determines whether or not a given finite subset X X of such a computable group generates a soluble group. When the group is soluble, the algorithm also determines the derived length of ⟨ X ⟩ \langle X\rangle . Finally, we give a solution of the membership problem for a particular family of finitely generated soluble subgroups of any computable subgroup of P L + ( I ) PL_+(I) .

We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group $F$ which is strictly well-ordered by the embeddability relation in type $\epsilon_0 +1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In fact we also obtain, for each $\alpha < \epsilon_0$, a finitely generated elementary amenable subgroup of $F$ whose EA-class is $\alpha + 2$. These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with $\mathbf{Z} + \mathbf{Z}$, $\mathbf{Z} \wr \mathbf{Z}$, and the Brin-Navas group $B$. We also give an example of a pair of finitely generated elementary amenable subgroups of $F$ with the property that neither is embeddable into the other.

Let 1 <= r < n be integers. We give a proof that the group Aut(X-n(N), sigma(n)) automorphisms of the one-sided shift on n letters embeds naturally as a subgroup J-Cn of outer automorphism group Out(G(n,r)) of the Higman-Thompson group G(n,r). From this, can represent the elements of Aut(X-n(N), sigma(n)) by finite state non-initial transducers admitting very strong synchronizing condition. Let H is an element of H-n and write vertical bar H vertical bar for the number of states of the minimal transducer representing H. We show that H can be written as a product of at most vertical bar H vertical bar 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 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 Aut(X-n(N), sigma(n)).

The Boone--Higman conjecture is that every recursively presented group with solvable word problem embeds in a finitely presented simple group. We discuss a brief history of this conjecture and work towards it. Along the way we describe some classes of finitely presented simple groups, and we briefly outline work of Belk, Bleak, Matucci, and Zaremsky showing that the broad class of hyperbolic groups embeds in a class of finitely presented simple groups.

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 Aut(X_n^ℕ, σ_n) by its centre embeds as a particular subgroup ℒ_n of the outer automorphism group Out(G_n,n-1) of G_n,n-1. It follows by a result of Ryan that we have the following central extension: ⟨σ_n⟩↪Aut(X_n^ℕ, σ_n) ↠ℒ_n where here, ⟨σ_n⟩≅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 Out(G_n,r) are centreless and have undecidable order problem. Note that the group Out(G_n,n-1) consists of finite transducers (combinatorial objects arising in automata theory), and elements of the group ℒ_n are easily characterised within 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 Aut(X_n^ℕ, σ_n), and we demonstrate how to compute products using this new representation.

We prove that every finitely-generated right-angled Artin group can be embedded into some Brin-Thompson group $nV$. It follows that many other groups can be embedded into some $nV$ (e.g., any finite extension of any of Haglund and Wise's special groups), and that various decision problems involving subgroups of $nV$ are unsolvable.