Finitely presented simple left-orderable groups in the landscape of Richard Thompson's 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.

Thompson's group V has a rich variety of subgroups, con- taining all finite groups, all finitely generated free groups and all finitely generated abelian groups, the finitary permutation group of a countable set, as well as many wreath products and other families of groups. Here, we describe some obstructions for a given group to be a subgroup of V. Thompson constructed a finitely presented group now known as V as an early example of a finitely presented infinite simple group. The group V contains a remarkable variety of subgroups, such as the finitary infinite per- mutation group S∞, and hence all (countable locally) finite groups, finitely generated free groups, finitely generated abelian groups, Houghton's groups, copies of Thompson's groups F, T and V , and many of their generalizations, such as the groups Gn,r constructed by Higman (9). Moreover, the class of subgroups of V is closed under direct products and restricted wreath products with finite or infinite cyclic top group. In this short survey, we summarize the development of properties of V focusing on those which prohibit various groups from occurring as subgroups of V. Thompson's group V has many descriptions. Here, we simply recall that V is the group of right-continuous bijections from the unit interval (0,1) to itself, which map dyadic rational numbers to dyadic rational numbers, which are differentiable except at finitely many dyadic rational numbers, and with slopes, when defined, integer powers of 2. The elements of this group can be described by reduced tree pair diagrams of the type (S,T,�) whereis a bijection between the leaves of the two finite rooted binary trees S and T. Higman (9) gave a different description of V , which he denoted as G2,1 in a family of groups generalizing V.

We construct a family of infinite simple groups that we call \emph{twisted Brin-Thompson groups}, generalizing Brin's higher-dimensional Thompson groups $sV$ ($s\in\mathbb{N}$). We use twisted Brin-Thompson groups to prove a variety of results regarding simple groups. For example, we prove that every finitely generated group embeds quasi-isometrically as a subgroup of a two-generated simple group, strengthening a result of Bridson. We also produce examples of simple groups that contain every $sV$ and hence every right-angled Artin group, including examples of type $\textrm{F}_\infty$ and a family of examples of type $\textrm{F}_{n-1}$ but not of type $\textrm{F}_n$, for arbitrary $n\in\mathbb{N}$. This provides the second known infinite family of simple groups distinguished by their finiteness properties.

On Thompson's Theorem

We determine all conjugacy classes of maximal local subgroups of Thompson's sporadic simple group, and all maximal non-local subgroups except those with socle isomorphic to one of five particular small simple groups.

We show that the abstract commensurator of Thompson's group F is composed of four building blocks: two isomorphism types of simple groups, the multiplicative group of the positive rationals and a cyclic group of order two. The main result establishes the simplicity of a certain group of piecewise linear homeomorphisms of the real line.

Thompson's sporadic group uniquely determined by the centralizer of a 2-central involution

Minimal permutation representation of Thompson's simple group

Elliptic curves and Thompson's sporadic simple group

We describe several exotic fusion systems related to the sporadic simple groups at odd primes. More generally, we classify saturated fusion systems supported on Sylow 3-subgroups of the Conway group $\textrm{Co}_1$ and the Thompson group $\textrm{F}_3$ , and a Sylow 5-subgroup of the Monster M, as well as a particular maximal subgroup of the latter two p-groups. This work is supported by computations in MAGMA.

A group G is said to be (l,m,n)-generated if it is a quotient group of the triangle group T(l,m,n)=〈x,y,z|xl=ym=zn=xyz=1〉. In 1993 J. Moori posed the question of finding all triples (l,m,n) such that a given non-abelian finite simple group is (l,m,n)-generated. In this paper we partially answer this question for the Thompson group Th. In fact we study (p,q,r)-generation, where p, q and r are distinct primes, and nX-complementary generations of the Thompson group Th.

We construct a finitely presented group with coNP-complete word problem, and a finitely generated simple group with coNP-complete word problem. These groups are represented as Thompson groups, hence as partial transformation groups of strings. The proof provides a simulation of combinational circuits by elements of the Thompson–Higman group G3,1.

The commencement of monstrous moonshine is a connection between the largest sporadic simple group---the monster---and complex elliptic curves. Here we explain how a closer look at this connection leads, via the Thompson group, to recently observed relationships between the non-monstrous sporadic simple group of O'Nan and certain families of elliptic curves defined over the rationals. We also describe umbral moonshine from this perspective.

The higher-dimensional Thompson groups $nV$ , for $n \geqslant 2$ , were introduced by Brin [‘Presentations of higher dimensional Thompson groups’, J. Algebra284 (2005), 520–558]. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in $nV$ and reflecting the self-similar structure of n-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.

A characterization of Thompson's sporadic simple group