Searcharxiv⌕ Search

arXiv subjects

James Hyde

Publications and source records attributed to James Hyde.

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↗
arXiv2026

Action graphs, semiconjugacy, and non-embedding in Thompson's group $V$

We prove a variety of results about subgroups of Thompson's group $V$. First we prove that every action graph of a finitely generated subgroup of $V$ acting on an orbit in Cantor space is quasi-isometric to a tree. Then we prove that for a broad class of groups of homeomorphisms of the real line, for example Thompson's group $F$, any action on the Cantor space via an embedding into Thompson's group $V$ must be semiconjugate to the standard action on the line. Finally, we use this to establish that many such groups cannot embed into $V$; in particular the Stein group $F_{2,3}$ cannot embed in $V$, answering a question of the third author.

math.GR↗
arXiv2026

On Ulam widths of finitely presented infinite simple groups

A fundamental notion in group theory, which originates in an article of Ulam and von Neumann from $1947$ is uniform simplicity. A group $G$ is said to be $n$-uniformly simple for $n \in \mathbf{N}$ if for every $f,g\in G\setminus \{id\}$, there is a product of no more than $n$ conjugates of $g$ and $g^{-1}$ that equals $f$. Then $G$ is uniformly simple if it is $n$-uniformly simple for some $n \in \mathbf{N}$, and we refer to the smallest such $n$ as the Ulam width, denoted as $\mathcal{R}(G)$. If $G$ is simple but not uniformly simple, one declares $\mathcal{R}(G)=\infty$. In this article, we construct for each $n\in \mathbf{N}$, a finitely presented infinite simple group $G$ such that $n<\mathcal{R}(G)<\infty$. These are the first such examples among the class of finitely presented infinite simple groups. For the class of finitely generated (but not finitely presentable) infinite simple groups, the existence of such examples was settled in the work of Muranov. However, this had remained open for the class of finitely presented infinite simple groups. Our examples are also of type $F_{\infty}$, which means that they are fundamental groups of aspherical CW complexes with finitely many cells in each dimension. Uniformly simple groups are in particular uniformly perfect: there is an $n\in \mathbf{N}$ such that every element of the group can be expressed as a product of at most $n$ commutators of elements in the group. We also show that the analogous notion of width for uniform perfection is unbounded for our family of finitely presented infinite simple groups. To our knowledge, this is also the first such family.

math.GR↗
arXiv2025

A piecewise linear homeomorphism of the circle which is periodic under renormalization

We demonstrate the existence of a piecewise linear homeomorphism $f$ of $\mathbb{R}/\mathbb{Z}$ which maps rationals to rationals, whose slopes are powers of $\frac{2}{3}$, and whose rotation number is $\sqrt{2}-1$. This is achieved by showing that a renormalization procedure becomes periodic when applied to $f$. Our construction gives a negative answer to a question of D. Calegari. When combined with work of the 2nd and 3rd authors, our result also shows that $F_{\frac{2}{3}}$ does not embed into $F$, where $F_{\frac{2}{3}}$ is the subgroup of the Stein-Thompson group $F_{2,3}$ consisting of those elements whose slopes are powers of $\frac{2}{3}$. Finally, we produce some evidence suggesting a positive answer to a variation of Calegari's question and record a number of computational observations.

math.GR↗
arXiv2025

Embeddings into highly transitive and mixed identity free groups

Given a countable group $G$, we develop a method to construct an overgroup $H$ that is finitely generated, highly transitive and mixed identity free. Our construction can be controlled to ensure that some fundamental group theoretic properties of $G$ are inherited by $H$, such as amenability or the property of not containing a nonabelian free group. The former provides a strong solution to a question of Hull and Osin, and the latter provides the first examples of nonamenable groups without free subgroups that are highly transitive and mixed identity free. Our examples also have a nontrivial amenable radical, answering a question of Arzhantseva.

math.GR↗
arXiv2025

Boone-Higman embeddings of $\mathrm{Aut}(F_n)$ and mapping class groups of punctured surfaces

We prove that the groups $\mathrm{Aut}(F_n)$ satisfy the Boone-Higman conjecture for all $n$, meaning each $\mathrm{Aut}(F_n)$ embeds in a finitely presented simple group. In fact, we prove that each $\mathrm{Aut}(F_n)$ satisfies the "permutational" Boone-Higman conjecture, which means the simple group in question can be taken to be a twisted Brin-Thompson group. A far-reaching consequence of our approach is that finitely presented twisted Brin-Thompson groups are universal among finitely presented simple groups that are highly transitive. This is evidence toward the Boone-Higman conjecture being equivalent to its permutational version. Proving the conjecture for $\mathrm{Aut}(F_n)$ also confirms the conjecture for all groups (virtually) embedding into some $\mathrm{Aut}(F_n)$, such as mapping class groups of non-closed surfaces, braid groups, loop braid groups, ribbon braid groups and certain Artin groups. This answers several questions of the first and fourth authors with Bleak and Matucci. Yet another consequence of our approach is that satisfying the permutational Boone-Higman conjecture is closed under free products.

math.GR↗
arXiv2024

Finite Germ Extensions

We prove finiteness properties for groups of homeomorphisms that have finitely many "singular points", and we describe the normal structure of such groups. As an application, we prove that every countable abelian group can be embedded into a finitely presented simple group, verifying the Boone-Higman conjecture for countable abelian groups. Indeed, we describe a specific 2-generated, $\mathrm{F}_\infty$ simple group $V\mathcal{A}$ of homeomorphisms of the Cantor set that contains every countable abelian group. As a second application, we prove that if $G$ is a bounded automata group then the associated Röver-Nekrashevych groups $V_{d,r}G$ have type $\mathrm{F}_\infty$, verifying a conjecture of Nekrashevych for a large class of contracting self-similar groups. Among others, this result applies to Röver-Nekrashevych groups associated to Gupta-Sidki groups and the basilica group.

math.GR↗
arXiv2023

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 $γ\in G$ in the pointwise-stabiliser of $\mathfrak{C}\backslash A$ with $Bγ\subseteq C$. 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 $nV$, Röver's group $V(Γ)$, and others of Nekrashevych's `simple groups of dynamical origin', and, the class is closed under various natural constructions).

math.GR↗
arXiv2022

Embedding $\mathbb{Q}$ into a Finitely Presented Group

We observe that the group of all lifts of elements of Thompson's group $T$ to the real line is finitely presented and contains the additive group $\mathbb{Q}$ of the rational numbers. This gives an explicit realization of the Higman embedding theorem for $\mathbb{Q}$, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.

math.GR↗
arXiv2021

Predicting Adversary Lateral Movement Patterns with Deep Learning

This paper develops a predictive model for which host, in an enterprise network, an adversary is likely to compromise next in the course of a campaign. Such a model might support dynamic monitoring or defenses. We generate data for this model using simulated networks, with hosts, users, and adversaries as first-class entities. We demonstrate the predictive accuracy of the model on out-of-sample simulated data, and validate the findings against data captured from a Red Team event on a live enterprise network

cs.CR↗
arXiv2021

Stabilizers in Higman-Thompson groups

We investigate stabilizers of finite sets of rational points in Cantor space for the Higman-Thompson groups $V_{n,r}$. We prove that the pointwise stabilizer is an iterated ascending HNN extension of $V_{n,q}$ for any $q\geq 1$. We also prove that the commutator subgroup of the pointwise stabilizer is simple, and we compute the abelianization. Finally, for each $n$ we classify such pointwise stabilizers up to isomorphism.

math.GR↗
arXiv2021

Subgroups of $\mathrm{PL}_+ I$ which do not embed into Thompson's group $F$

We will give a general criterion - the existence of an $F$-obstruction - for showing that a subgroup of $\mathrm{PL}_+ I$ does not embed into Thompson's group $F$. An immediate consequence is that Cleary's "golden ratio" group $F_τ$ does not embed into $F$. Our results also yield a new proof that Stein's groups $F_{p,q}$ do not embed into $F$, a result first established by Lodha using his theory of coherent actions. We develop the basic theory of $F$-obstructions and show that they exhibit certain rigidity phenomena of independent interest. In the course of establishing the main result of the paper, we prove a dichotomy theorem for subgroups of $\mathrm{PL}_+ I$. In addition to playing a central role in our proof, it is strong enough to imply both Rubin's Reconstruction Theorem restricted to the class of subgroups of $\mathrm{PL}_+ I$ and also Brin's Ubiquity Theorem.

math.GR↗
arXiv2021

Two new families of finitely generated simple groups of homeomorphisms of the real line

The goal of this article is to exhibit two new families of finitely generated simple groups of homeomorphisms of $\mathbf{R}$. These families are strikingly different from existing families owing to the nature of their actions on $\mathbf{R}$, and exhibit surprising algebraic and dynamical features. In particular, one construction provides the first examples of finitely generated simple groups of homeomorphisms of the real line which also admit a minimal action by homeomorphisms on the circle. This provides new examples of finitely generated simple groups with infinite commutator width, and the first such left orderable examples. Another construction provides the first examples of finitely generated simple left orderable groups that admit minimal actions by homeomorphisms on the torus.

math.GR↗
arXiv2019

Sets of universal sequences for the symmetric group and analogous semigroups

A universal sequence for a group or semigroup $S$ is a sequence of words $w_1, w_2, \ldots$ such that for any sequence $s_1, s_2, \ldots\in S$, the equations $w_n = s_n$, $n\in \mathbb{N}$, can be solved simultaneously in $S$. For example, Galvin showed that the sequence $(a^{-1}(a^nba^{-n})b^{-1}(a^nb^{-1}a^{-n})ba)_{n\in\mathbb{N}}$ is universal for the symmetric group Sym$(X)$ when $X$ is infinite, and Sierpiński showed that $(a ^ 2 b ^ 3 (abab ^ 3) ^ {n + 1} ab ^ 2 ab ^ 3)_{n\in \mathbb{N}}$ is universal for the monoid $X ^ X$ of functions from the infinite set $X$ to itself. In this paper, we show that under some conditions, the set of universal sequences for the symmetric group on an infinite set $X$ is independent of the cardinality of $X$. More precisely, we show that if $Y$ is any set such that $|Y| \geq |X|$, then every universal sequence for Sym$(X)$ is also universal for Sym$(Y)$. If $|X| > 2 ^ {\aleph_0}$, then the converse also holds. It is shown that an analogue of this theorem holds in the context of inverse semigroups, where the role of the symmetric group is played by the symmetric inverse monoid. In the general context of semigroups, the full transformation monoid $X ^ X$ is the natural analogue of the symmetric group and the symmetric inverse monoid. If $X$ and $Y$ are arbitrary infinite sets, then it is an open question as to whether or not every sequence that is universal for $X ^ X$ is also universal for $Y ^ Y$. However, we obtain a sufficient condition for a sequence to be universal for $X ^ X$ which does not depend on the cardinality of $X$. A large class of sequences satisfy this condition, and hence are universal for $X ^ X$ for every infinite set $X$.

math.GR↗
arXiv2019

Finitely generated infinite simple groups of homeomorphisms of the real line

We construct examples of finitely generated infinite simple groups of homeomorphisms of the real line. Equivalently, these are examples of finitely generated simple left (or right) orderable groups. This answers a well known open question of Rhemtulla from 1980 concerning the existence of such groups. In fact, our construction provides a family of continuum many isomorphism types of groups with these properties.

math.GR↗
arXiv2019

The Group of Boundary Fixing Homeomorphisms of the Disc is Not Left-Orderable

A left-order on a group $G$ is a total order $<$ on $G$ such that for any $f$, $g$ and $h$ in $G$ we have $f < g \Leftrightarrow hf < hg$. We construct a finitely generated subgroup $G$ of $\operatorname{Homeo} (I^2;δI^2)$, the group of those homeomorphisms of the disc that fix the boundary pointwise, and show $G$ does not admit a left-order. Since any left-order on $\operatorname{Homeo} (I^2;δI^2)$ would restrict to a left-order on $G$ this shows that $\operatorname{Homeo} (I^2;δI^2)$ does not admit a left-order. Since $\operatorname{Homeo} (I;δI)$ admits a left-order it follows that neither $G$ nor $\operatorname{Homeo} (I^2;δI^2)$ embed in $\operatorname{Homeo} (I;δI)$.

math.GR↗
arXiv2019

Uniformly perfect finitely generated simple left orderable groups

We show that the finitely generated simple left orderable groups $G_ρ$ constructed by the first two authors in arXiv:1807.06478 are uniformly perfect - each element in the group can be expressed as a product of three commutators of elements in the group. This implies that the group does not admit any homogeneous quasimorphism. Moreover, any nontrivial action of the group on the circle, which lifts to an action on the real line, admits a fixed point. Most strikingly, it follows that the groups are examples of left orderable monsters, which means that any faithful action on the real line without a global fixed point is globally contracting. This answers Question 4 from the 2018 ICM proceedings article of the third author. (This question has also been answered simultaneously and independently, using completely different methods, by Matte Bon and Triestino in arXiv:1811.12256.) To prove our results, we provide a certain characterisation of elements of the group $G_ρ$ which is a useful new tool in the study of these examples.

math.GR↗