arXiv subjects

Collin Bleak

Publications and source records attributed to Collin Bleak.

At least 19 recordsLinked to original sources

arXiv2026

Generating simple vigorous groups

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 $m \geq 2$ and $n \geq 3$, $G$ has a minimal generating set of size $k$ for all $k \geq 2$, 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 $V$.

math.GR↗

arXiv2025

On epiC groups over language class 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 $X \subseteq G$ in the class C such that the image of L under the evaluation map is $G \setminus \{1_G\}$. 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.

math.GR↗

arXiv2024

The Maximality of $T$ in Thompson's group $V$

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 Cantor space $\mathfrak{C}$.

math.GR↗

arXiv2023

Hyperbolic groups satisfy the Boone-Higman conjecture

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.

math.GR↗

arXiv2023

Progress around the Boone-Higman Conjecture

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.

math.GR↗

arXiv2023

Embedding hyperbolic groups into finitely presented infinite simple groups

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.

math.GR↗

arXiv2023

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 $\psi$ in the group of automorphisms of the one-sided shift $\operatorname{Aut}(\{0,1,\ldots,n-1\}^{\mathbb{N}}, \sigma_{n})$ so that all points of $\{0,1,\ldots,n-1\}^{\mathbb{N}}$ have orbits of length $n$ under $\psi$ and $\psi$ 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 $\psi$ 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↗

arXiv2022

Type systems and maximal subgroups of Thompson's group $V$

We introduce the concept of a type system~$\Part$, that is, a partition on the set of finite words over the alphabet~$\{0,1\}$ compatible with the partial action of Thompson's group~$V$, and associate a subgroup~$\Stab{V}{\Part}$ of~$V$. 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~$V$. We also find an uncountable family of pairwise non-isomorphic maximal subgroups of~$V$. 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 $V$ of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of $V$ (both related to primitivity) are each satisfied only by $V$ itself, giving new ways to recognise when a subgroup of $V$ is not actually proper.

math.GR↗

arXiv2022

Thompson's group $T$ is $\frac{3}{2}$-generated

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 $\langle \alpha, \zeta^\gamma \rangle = T$.

math.GR↗

arXiv2020

Sufficient conditions for a group of homeomorphisms of the Cantor set to be two-generated

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 $nV$, R\"{o}ver's group $V(\Gamma)$, and others of Nekrashevych's `simple groups of dynamical origin', and, the class is closed under various natural constructions).

math.GR↗

arXiv2020

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 $\mathrm{Aut}(X_n^{\mathbb{N}}, \sigma_{n})$ 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: $$\langle \sigma_{n}\rangle \hookrightarrow \mathrm{Aut}(X_n^{\mathbb{N}}, \sigma_{n}) \twoheadrightarrow \mathcal{L}_{n}$$ where here, $\langle \sigma_{n} \rangle \cong \mathbb{Z}$. 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 $\mathrm{Aut}(X_n^{\mathbb{N}}, \sigma_{n})$, and we demonstrate how to compute products using this new representation.

math.GR↗

arXiv2020

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 $\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, \sigma_{n}})$ 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 $\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, \sigma_{n}})$ 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 $\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, \sigma_{n}})$.

math.GR↗

arXiv2020

Some embeddings between symmetric R. Thompson groups

Let $m\leq n\in \mathbb{N}$, and $G\leq S_m$ and $H\leq S_n$. 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 $V_n(H)$ in $V_m(G)$ via topological conjugation. With the same modular condition we also generalise a purely algebraic construction of Birget from 2019 to find a group $H\leq S_m$ and an embedding of $V_m(G)$ in $V_n(H)$.

math.GR↗

arXiv2017

Complexity among the finitely generated subgroups of Thompson's group

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.

math.GR↗

arXiv2017

Rational embeddings of hyperbolic groups

We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanski\u{i}. The proof involves assigning a system of binary addresses to points in the Gromov boundary of $G$, and proving that elements of $G$ act on these addresses by transducers. These addresses derive from a certain self-similar tree of subsets of $G$, whose boundary is naturally homeomorphic to the horofunction boundary of $G$.

math.GR↗

arXiv2017

Groups of fast homeomorphisms of the interval and the ping-pong argument

We adapt the Ping-Pong Lemma, which historically was used to study free products of groups, to the setting of the homeomorphism group of the unit interval. As a consequence, we isolate a large class of generating sets for subgroups of $\mathrm{Homeo}_+(I)$ for which certain finite dynamical data can be used to determine the marked isomorphism type of the groups which they generate. As a corollary, we will obtain a criteria for embedding subgroups of $\mathrm{Homeo}_+(I)$ into Richard Thompson's group $F$. In particular, every member of our class of generating sets generates a group which embeds into $F$ and in particular is not a free product. An analogous abstract theory is also developed for groups of permutations of an infinite set.

math.GR↗

arXiv2016

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 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.

math.GR↗

arXiv2016

Determining solubility for finitely generated groups of PL homeomorphisms

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

math.GR↗