arXiv subjects

Charles Garnet Cox

Publications and source records attributed to Charles Garnet Cox.

10 recordsLinked to original sources

arXiv2026

A note on the $R_\infty$ property for groups $\mathrm{FAlt}(X)\leqslant G\leqslant \mathrm{Sym}(X)$

Given a set $X$, the group $\mathrm{Sym}(X)$ consists of all bijections from $X$ to $X$, and $\mathrm{FSym}(X)$ is the subgroup of maps with finite support i.e. those that move only finitely many points in $X$. We describe the automorphism structure of groups $\mathrm{FSym}(X)\le G\le \mathrm{Sym}(X)$ and use this to state some conditions on $G$ for it to have the $R_\infty$ property. Our main results are that if $G$ is infinite, torsion, and $\mathrm{FSym}(X)\le G\le \mathrm{Sym}(X)$, then it has the $R_\infty$ property. Also, if $G$ is infinite and residually finite, then there is a set $X$ such that $G$ acts faithfully on $X$ and, using this action, $\langle G, \mathrm{FSym}(X)\rangle$ has the $R_\infty$ property. Finally we have a result for the Houghton groups, which are a family of groups we denote $H_n$, where $n \in \mathbb{N}$. We show that, given any $n\in \mathbb{N}$, any group commensurable to $H_n$ has the $R_\infty$ property.

math.GR↗

arXiv2026

Finiteness properties of Subgroups of Houghton Groups of full Hirsch length

In the 1980's K.S. Brown proved that the Houghton group $H_n$ is of type $\operatorname{F}_{n-1}$ but not $\operatorname{FP}_n$. We show that, provided $n\ge3$, the same conclusion holds for all subgroups $G$ of $H_n$ that are 'large' in the sense that there is an epimorphism $G\twoheadrightarrow\mathbb{Z}^{n-1}$. Our research leads naturally to the study of generalised permutational wreath products in which the base of the wreath product is a direct product of finite groups which are allowed to vary in isomorphism type from one orbit to another. Such generalised wreath products arise naturally amongst the large subgroups of Houghton groups and are accommodated by a generalised Jordan--Wielandt theorem.

math.GR↗

arXiv2025

Invariable generation of certain branch groups

Let $G$ be a group. Then $S\subseteq G$ is an invariable generating set of $G$ if every subset $S'$ obtained from $S$ by replacing each element with a conjugate is also a generating set of $G$. We investigate invariable generation among key examples of branch groups. In particular, we prove that all generating sets of the torsion Grigorchuk groups, of the branch Grigorchuk-Gupta-Sidki groups and of the torsion multi-EGS groups (which are natural generalisations of the Grigorchuk-Gupta-Sidki groups) are invariable generating sets. Furthermore, for the first Grigorchuk group and the torsion Grigorchuk-Gupta-Sidki groups, every finitely generated subgroup has a finite invariable generating set. Our results apply to finitely generated groups in $\mathcal{MN}$, the class of groups whose maximal subgroups are all normal. We then obtain that any $2$-generated group in $\mathcal{MN}$ is almost $\frac{3}{2}$-generated, and end by applying this observation to generating graphs.

math.GR↗

arXiv2023

On the finite index subgroups of Houghton's groups

An erratum has been added to resolve an issue raised by Professor Derek Holt. This appears after the original paper, and also includes two new results. Original abstract: Houghton's groups $H_2, H_3, \ldots$ are certain infinite permutation groups acting on a countably infinite set; they have been studied, among other things, for their finiteness properties. In this note we describe all of the finite index subgroups of each Houghton group, and their isomorphism types. Using the standard notation that $d(G)$ denotes the minimal size of a generating set for $G$ we then show, for each $n\in \{2, 3,\ldots\}$ and $U$ of finite index in $H_n$, that $d(U)\in\{d(H_n), d(H_n)+1\}$ and characterise when each of these cases occurs.

math.GR↗

arXiv2021

On the spread of infinite groups

A group is $\frac32$-generated if every non-trivial element is part of a generating pair. In 2019, Donoven and Harper showed that many Thompson groups are $\frac32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups of the Houghton group $\text{FSym}(\mathbb{Z})\rtimes\mathbb{Z}$. We then show that the first two groups in our family are $\frac32$-generated, and investigate the related notion of spread for these groups. We are able to show that they have finite spread which is greater than 2. These are therefore the first infinite groups to be shown to have finite positive spread, and the first to be shown to have spread greater than 2 (other than $\mathbb{Z}$ and the Tarski monsters, which have infinite spread).

math.GR↗

arXiv2020

Invariable generation and wreath products

Invariable generation is a topic that has predominantly been studied for finite groups. In 2014, Kantor, Lubotzky, and Shalev produced extensive tools for investigating invariable generation for infinite groups. Since their paper, various authors have investigated the property for particular infinite groups or families of infinite groups. A group is invariably generated by a subset $S$ if replacing each element of $S$ with any of its conjugates still results in a generating set for $G$. In this paper we investigate how this property behaves with respect to wreath products. Our main work is to deal with the case where the base of $G\wr_X H$ is not invariably generated. We see both positive and negative results here depending on $H$ and its action on $X$.

math.GR↗

arXiv2020

Invariable generation and the Houghton groups

The Houghton groups $H_1, H_2, \ldots$ are a family of infinite groups. In 1975 Wiegold showed that $H_3$ was invariably generated (IG) but $H_1\le H_3$ was not. A natural question is then whether the groups $H_2, H_3, \ldots$ are all IG. Wiegold also ends by saying that, in the examples he had found of an IG group with a subgroup that is not IG, the subgroup was never of finite index. Another natural question is then whether there is a subgroup of finite index in $H_3$ that is not IG. In this note we prove, for each $n\in \{2, 3, \ldots\}$, that $H_n$ and all of its finite index subgroups are IG. The independent work of Minasyan and Goffer-Lazarovich in June 2020 frames this note quite nicely: they showed that an IG group can have a finite index subgroup that is not IG.

math.GR↗

arXiv2018

The degree of commutativity and lamplighter groups

The degree of commutativity of a group $G$ measures the probability of choosing two elements in $G$ which commute. There are many results studying this for finite groups. In [AMV17], this was generalised to infinite groups. In this note, we compute the degree of commutativity for wreath products of the form $\mathbb{Z}\wr \mathbb{Z}$ and $F\wr \mathbb{Z}$ where $F$ is any finite group.

math.GR↗

arXiv2017

The conjugacy ratio of groups

In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is $0$ for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group.

math.GR↗

arXiv2017

Twisted Conjugacy in Houghton's groups

For a fixed $n\ge2$, the Houghton group $H_n$ consists of bijections of $X_n=\{1,\ldots,n\} \times \mathbb{N}$ that are `eventually translations' of each copy of $\mathbb{N}$. The Houghton groups have been shown to have solvable conjugacy problem. In general solvable conjugacy problem does not imply that all finite extensions and finite index subgroups have solvable conjugacy problem. Our main theorem is that a stronger result holds: for any $n\ge2$ and any group $G$ commensurable to $H_n$, $G$ has solvable conjugacy problem.

math.GR↗