Two new avatars of moonshine for the Thompson group

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 give a unified solution to the conjugacy problem for Thompson's groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson's groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson's groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.

Using ideas from Jones, lattice gauge theory and loop quantum gravity, we construct 1+1-dimensional gauge theories on a spacetime cylinder. Given a separable compact group $G$, we construct localized time-zero fields on the spatial torus as a net of C*-algebras together with an action of the gauge group that is an infinite product of $G$ over the dyadic rationals and, using a recent machinery of Jones, an action of Thompson's group $T$ as a replacement of the spatial diffeomorphism group. Adding a family of probability measures on the unitary dual of $G$ we construct a state and obtain a net of von Neumann algebras carrying a state-preserving gauge group action. For abelian $G$, we provide a very explicit description of our algebras. For a single measure on the dual of $G$, we have a state-preserving action of Thompson's group and semi-finite von Neumann algebras. For $G=\mathbf{S}$ the circle group together with a certain family of heat-kernel states providing the measures, we obtain hyperfinite type III factors with a normal faithful state providing a nontrivial time evolution via Tomita-Takesaki theory (KMS condition). In the latter case, we additionally have a non-singular action of the group of rotations with dyadic angles, as a subgroup of Thompson's group $T$, for geometrically motivated choices of families of heat-kernel states.

We describe some properties of braided generalizations of Thompson's groups, introduced by Brin and Dehornoy. We give slightly different characterizations of the braided Thompson's groups $BV$ and $\widehat{BV}$ which lead to natural presentations which emphasize one of their subgroup-containment properties. We consider pure braided versions of Thompson's group $F$. These groups, $BF$ and $\widehat{BF}$, are subgroups of the braided versions of Thompson's group $V$. Unlike $V$, elements of $F$ are order-preserving self-maps of the interval and we use pure braids together with elements of $F$ thus again preserving order. We define these pure braided groups, give normal forms for elements, and construct infinite and finite presentations of these groups.

Recall that a group is said to be ‐generated if every nontrivial element of belongs to a generating pair of . Thompson's group was proved to be ‐generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented noncyclic ‐generated group. Recently, Bleak, Harper, and Skipper proved that Thompson's group is also ‐generated. In this paper, we prove that Thompson's group is “almost” ‐generated in the sense that every element of whose image in the abelianization forms part of a generating pair of is part of a generating pair of . We also prove that for every nontrivial element , there is an element such that the subgroup contains the derived subgroup of . Moreover, if does not belong to the derived subgroup of , then there is an element such that has finite index in .

Braided Thompson's groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups B n , the finitary braid group B ∞ and Thompson's group F as subgroups. We describe some of the metric properties of braided Thompson's groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams used to represent elements.

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.

The main result of this article is that any braided (resp. annular, planar) diagram group D splits as a short exact sequence 1 \to R \to D \to S \to 1 where R is a subgroup of some right-angled Artin group and S a subgroup of Thompson's group V (resp. T , F ). As an application, we show that several braided diagram groups embed into Thompson's group V , including Higman's groups V_{n,r} , groups of quasi-automorphisms QV_{n,r,p} , and generalised Houghton's groups H_{n,p} .

The length-based approach is a heuristic for solving randomly generated equations in groups which possess a reasonably behaved length function. We describe several improvements of the previously suggested length-based algorithms, that make them applicable to Thompson's group with significant success rates. In particular, this shows that the Shpilrain-Ushakov public key cryptosystem based on Thompson's group is insecure, and suggests that no practical public key cryptosystem based on this group can be secure.

Cone types and geodesic languages for lamplighter groups and Thompson's group F

This chapter considers the Thompson's group F. Thompson's group F exhibits several behaviors that appear paradoxical. For example: F is finitely presented and contains a copy of F x F, indicating that F contains the direct sum of infinitely many copies of F. In addition, F has exponential growth but contains no free groups of rank 2. After providing an overview of the analytic definition and basic properties of the Thompson's group, the chapter introduces a combinatorial definition of F and two group presentations for F, an infinite one and a finite one. It also explores the subgroups, quotients, endomorphisms, and group action of F before concluding with an analysis of several geometric properties of F such as word length, distortion, dead ends, and growth. The discussion includes exercises and research projects.

Lehnert and Schweitzer show in [20] that R. Thompson's group $V$ is a co-context-free ($co\mathcal{CF}$) group, thus implying that all of its finitely generated subgroups are also $co\mathcal{CF}$ groups. Also, Lehnert shows in his thesis that $V$ embeds inside the $co\mathcal{CF}$ group $\mathrm{QAut}(\mathcal{T}_{2,c})$, which is a group of particular bijections on the vertices of an infinite binary $2$-edge-colored tree, and he conjectures that $\mathrm{QAut}(\mathcal{T}_{2,c})$ is a universal $co\mathcal{CF}$ group. We show that $\mathrm{QAut}(\mathcal{T}_{2,c})$ embeds into $V$, and thus obtain a new form for Lehnert's conjecture. Following up on these ideas, we begin work to build a representation theory into R. Thompson's group $V$. In particular we classify precisely which Baumslag-Solitar groups embed into $V$.

Brown has defined the generalised Thompson's group $F_n$, $T_n$, where $n$ is an integer at least $2$ and Thompson's groups $F= F_2$ and $T =T_2$ in the 80's. Burillo, Cleary and Stein have found that there is a quasi-isometric embedding from $F_n$ to $F_m$ where $n$ and $m$ are positive integers at least 2. We show that there is a quasi-isometric embedding from $T_n$ to $T_2$ for any $n \geq 2$ and no embeddings from $T_2$ to $T_n$ for $n \geq 3$.

Results in [Formula: see text] algebras, of Matte Bon and Le Boudec, and of Haagerup and Olesen, apply to the R. Thompson groups [Formula: see text]. These results together show that [Formula: see text] is non-amenable if and only if [Formula: see text] has a simple reduced [Formula: see text]-algebra. In further investigations into the structure of [Formula: see text]-algebras, Breuillard, Kalantar, Kennedy, and Ozawa introduce the notion of a normalish subgroup of a group [Formula: see text]. They show that if a group [Formula: see text] admits no non-trivial finite normal subgroups and no normalish amenable subgroups then it has a simple reduced [Formula: see text]-algebra. Our chief result concerns the R. Thompson groups [Formula: see text]; we show that there is an elementary amenable group [Formula: see text] [where here, [Formula: see text]] with [Formula: see text] normalish in [Formula: see text]. The proof given uses a natural partial action of the group [Formula: see text] on a regular language determined by a synchronising automaton in order to verify a certain stability condition: once again highlighting the existence of interesting intersections of the theory of [Formula: see text] with various forms of formal language theory.

Thompson's group F is not almost convex