arXiv subjects

Casey Donoven

Publications and source records attributed to Casey Donoven.

7 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↗

arXiv2020

Infinite $\frac{3}{2}$-generated groups

Every finite simple group can be generated by two elements, and Guralnick and Kantor proved that, moreover, every nontrivial element is contained in a generating pair. Groups with this property are said to be $\frac{3}{2}$-generated. Thompson's group $V$ was the first finitely presented infinite simple group to be discovered. The Higman--Thompson groups $V_n$ and the Brin--Thompson groups $mV$ are two families of finitely presented groups that generalise $V$. In this paper, we prove that all of the groups $V_n$, $V_n'$ and $mV$ are $\frac{3}{2}$-generated. As far as the authors are aware, the only previously known examples of infinite noncyclic $\frac{3}{2}$-generated groups are the pathological Tarski monsters. We conclude with several open questions motivated by our results.

math.GR↗

arXiv2020

Groups that are the union of two semigroups have left-orderable quotients

In this article, we show that a group $G$ is the union of two proper subsemigroups if and only if $G$ has a nontrivial left-orderable quotient. Furthermore, if $G$ is the union of two proper semigroups, then there exists a minimum normal subgroup $N\unlhd G$ for which $G/N$ is left-orderable and nontrivial.

math.GR↗

arXiv2020

Finite Coverings of Semigroups and Related Structures

For a semigroup $S$, the covering number of $S$ with respect to semigroups, $σ_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all $n\geq 2$, there exists an inverse semigroup with covering number $n$, similar to the case of loops. Finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well.

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↗

arXiv2015

Codimension formulae for the intersection of fractal subsets of Cantor spaces

We examine the dimensions of the intersection of a subset $E$ of an $m$-ary Cantor space $\mathcal{C}^m$ with the image of a subset $F$ under a random isometry with respect to a natural metric. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. The dimensions of the intersections are typically $\max\{\dim E +\dim F -\dim \mathcal{C}^m, 0\}$, akin to other codimension theorems. The upper estimates come from the expected sizes of coverings, whilst the lower estimate is more intricate, using martingales to define a random measure on the intersection to facilitate a potential theoretic argument.

math.MG↗

arXiv2014

Some isomorphism results for Thompson like groups $V_n(G)$

We consider a class of groups $V_n(G)$ which are supergroups of the Higman-Thompson groups $V_n$. These groups fit in a framework of Elizabeth Scott for generating infinite virtually simple groups, and the groups we study in particular are initially introduced by Farley and Hughes. The group $V_n(G)$ is the result one obtains by taking the $V_n$ generators and adding a tree automorphism for each generator of a subgroup $G$ of the symmetric group on $n$ letters, where the new generators each permute the child leaves of a specific vertex $α$ of the infinite rooted $n$-ary tree according to the permutation they represent, and then they iterate this permutation again at each vertex which is a descendent of $α$. Farley and Hughes show that $V_n(G)$ is not isomorphic to $V_n$ when $G$ fails to act freely on the points $\{1,2,...,n\}$, and expect further non-isomorphism results in the other cases. We show the perhaps surprising result that if $G$ does act freely, then $V_n(G)\cong V_n$. We also generalise these results and produce some examples of even more isomorphisms amongst groups in the family $V_n(G)$. Essential tools in the above work are a study of the dynamics of the action of elements of $V_n(G)$ on Cantor space, Rubin's Theorem, and transducers from Grigorchuk, Nekrashevych, and Suschanskiĭ's rational group on the $n$-ary alphabet.

math.GR↗