Thompson's Group Research Articles

On Jones’ subgroup of R. Thompson’s group T

Jones introduced unitary representations for the Thompson groups [Formula: see text] and [Formula: see text] from a given subfactor planar algebra. Some interesting subgroups arise as the stabilizer of certain vector, in particular the Jones subgroups [Formula: see text] and [Formula: see text]. Golan and Sapir studied [Formula: see text] and identified it as a copy of the Thompson group [Formula: see text]. In this paper, we completely describe [Formula: see text] and show that [Formula: see text] coincides with its commensurator in [Formula: see text], implying that the corresponding unitary representation is irreducible. We also generalize the notion of the Stallings 2-core for diagram groups to [Formula: see text], showing that [Formula: see text] and [Formula: see text] are not isomorphic, but as annular diagram groups they have very similar presentations.

Presentations of generalisations of Thompson’s group V

We consider generalisations of Thompson's group $V$, denoted by $V_r(\Sigma)$, which also include the groups of Higman, Stein and Brin. It was shown by the authors in [20] that under some mild conditions these groups and centralisers of their finite subgroups are of type $\mathrm{F}_\infty$. Under more general conditions we show that the groups $V_r(\Sigma)$ are finitely generated and, under the mild conditions mentioned above, we see that they are finitely presented and give a recipe to find explicit presentations. For the centralisers of finite subgroups we find a suitable infinite presentation and then apply a general procedure to shorten this presentation. In the appendix, we give a proof of this general shortening procedure.

The Homflypt polynomial and the oriented Thompson group

We show how to construct unitary representations of the oriented Thompson group \vec F from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of \vec F .

Embeddings of right-angled Artin groups into higher-dimensional Thompson groups

In this paper, we show that every right-angled Artin group embeds into the [Formula: see text]-dimensional Thompson group [Formula: see text] for some [Formula: see text], by constructing a set of embeddings. This is an improvement of the previous result of Belk, Bleak and Matucci, which gives a different set of embeddings from right-angled Artin groups into higher-dimensional Thompson groups. Compared to their construction, the dimension [Formula: see text] of the target group is always smaller with our construction.

Quasi-automorphisms of the infinite rooted 2-edge-coloured binary tree

We study the group QV , the self-maps of the infinite 2 -edge coloured binary tree which preserve the edge and colour relations at cofinitely many locations. We introduce related groups QF , QT , \widetilde{Q}T , and \widetilde{Q}V , prove that QF , \widetilde{Q}T , and \widetilde{Q}V are of type \mathrm F_\infty , and calculate finite presentations for them. We calculate the normal subgroup structure of all 5 groups, the Bieri–Neumann–Strebel–Renz invariants of QF , and discuss the relationship of all 5 groups with other generalisations of Thompson's groups.

Quasiautomorphism groups of type F∞

The groups QF, QT, and QV are groups of quasi-automorphisms of the infinite binary tree. Their names indicate a similarity with Thompson's well-known groups F, T, and V. We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type F-infinity. Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on CAT(0) cubical complexes.

Quasisymmetries of the Basilica and the Thompson Group

We give a description of the group of all quasisymmetric self-maps of the Julia set of f(z) = z2−1 that have orientation preserving homeomorphic extensions to the whole plane. More precisely, we prove that this group is the uniform closure of the group generated by the Thompson group of the unit circle and an inversion. Moreover, this result is quantitative in the sense that distortions of the approximating maps are uniformly controlled by the distortion of the given map.

Diffeomorphism groups of tame Cantor sets and Thompson-like groups

The group of ${\mathcal{C}}^{1}$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations $nV$ of Thompson’s group $V$ arise when we consider products of central ternary Cantor sets. We derive that the ${\mathcal{C}}^{2}$-smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

Citrullination Inhibits SERPIN Activity

The Protein Arginine Deiminases (PADs) hydrolyze arginine residues to form citrulline. This post‐translational modification, termed citrullination, is upregulated in numerous cancers and autoimmune disorders including Rheumatoid Arthritis (RA), lupus, and Alzheimer's disease. While these enzymes are important regulators of gene transcription, the full spectrum of biological activities is relatively underexplored. Recently, the Thompson group reported the development of a suite of probes to detect and isolate citrullinated proteins. These probes are premised on the chemo‐selective reaction of citrulline with phenylglyoxals under acidic conditions. Herein, I will discuss our development of these probes and their use to uncover the RA citrullinome. Specifically, these probes were used to identify the citrullinated proteins present in RA serum, RA synovial fluid, and RA synovial tissue. The proteins identified from these analyses include >150 proteins that were previously not known to be citrullinated. Moreover, these proteins include numerous metabolic enzymes and serine protease inhibitors (SERPINs). Citrullination inactivates the activity of a subset of these proteins, including nicotinamide N‐methyltransferase and numerous P1‐arginine containing SERPINs (e.g., C1 inhibitor, anti‐thrombin, and anti‐plasmin). The ability to inhibit SERPIN activity is highly significant because it suggests PAD release within the RA synovium leads to SERPIN citrullination with a consequent increase in protease activity, thereby activating multiple processes including tissue remodeling, altered blood coagulation, and the activation of the complement cascade.Support or Funding InformationThis work was supported in part by NIH grant GM109767 to P.R.T.This abstract is from the Experimental Biology 2018 Meeting. There is no full text article associated with this abstract published in The FASEB Journal.

Thompson groups for systems of groups, and their finiteness properties

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F , V , V_{\mathrm{br}} and F_{\mathrm{br}} arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pure braid groups. We give new examples of families of groups that admit a cloning system and study how the finiteness properties of the resulting generalized Thompson group depend on those of the original groups. The main new examples here include upper triangular matrix groups, mock reflection groups, and loop braid groups. For generalized Thompson groups of upper triangular matrix groups over rings of S -integers of global function fields, we develop new methods for (dis-)proving finiteness properties, and show that the finiteness length of the generalized Thompson group is exactly the limit inferior of the finiteness lengths of the groups in the family.

On normal subgroups of the braided Thompson groups

We inspect the normal subgroup structure of the braided Thompson groups V_{\mathrm{br}} and F_{\mathrm{br}} . We prove that every proper normal subgroup of V_{\mathrm{br}} lies in the kernel of the natural quotient V_{\mathrm{br}} \twoheadrightarrow V , and we exhibit some families of interesting such normal subgroups. For F_{\mathrm{br}} , we prove that for any normal subgroup N of F_{\mathrm{br}} , either N is contained in the kernel of F_{\mathrm{br}} \twoheadrightarrow F , or else N contains [F_{\mathrm{br}},F_{\mathrm{br}}] . We also compute the Bieri–Neumann–Strebel invariant \Sigma^1(F_{\mathrm{br}}) , which is a useful tool for understanding normal subgroups containing the commutator subgroup.

Orderings of Witzel–Zaremsky–Thompson groups

ABSTRACTWe prove the orderability of the Witzel-Zaremsky-Thompson group for a direct system of orderable groups under a certain compatibility assumption.

Moonshine for all finite groups

In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O’Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group G, we give constructions of infinitely many graded infinite-dimensional $$\mathbb {C}[G]$$ -modules where the McKay–Thompson series for a conjugacy class [g] is a weakly holomorphic modular function properly on $$\varGamma _0({{\mathrm{ord}}}(g))$$ . As there are only finitely many normalized Hauptmoduln, groups whose McKay–Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level $${{\mathrm{ord}}}(g)$$ as the graded trace functions for any conjugacy class [g], and completely classify the finite abelian groups with this property. In particular, these include $$(\mathbb {Z}/ 5 \mathbb {Z})^5$$ and $$(\mathbb {Z}/ 7 \mathbb {Z})^4$$ , which are not subgroups of the Monster.

On the stabilizers of finite sets of numbers in the R. Thompson group $F$

We study subgroups $H_U$ of the R. Thompson group $F$ which are stabilizers of finite sets $U$ of numbers in the interval $(0,1)$. We describe the algebraic structure of $H_U$ and prove that the stabilizer $H_U$ is finitely generated if and only if $U$ consists of rational numbers. We also show that such subgroups are isomorphic surprisingly often. In particular, we prove that if finite sets $U\subset [0,1]$ and $V\subset [0,1]$ consist of rational numbers which are not finite binary fractions, and $|U|=|V|$, then the stabilizers of $U$ and $V$ are isomorphic. In fact these subgroups are conjugate inside a subgroup $\bar F<\Homeo([0,1])$ which is the completion of $F$ with respect to what we call the Hamming metric on $F$. Moreover the conjugator can be found in a certain subgroup $\F < \bar F$ which consists of possibly infinite tree-diagrams with finitely many infinite branches. We also show that the group $\F$ is non-amenable.

From skein theory to presentations for Thompson group

Jones introduced some unitary representations of Thompson group F constructed from a given subfactor planar algebra, and all unoriented links arise as matrix coefficients of these representations. Moreover, all oriented links arise as matrix coefficients of a subgroup F→ which is the stabilizer of a certain vector. Later Golan and Sapir determined the subgroup F→ and showed many interesting properties. In this paper, we investigate into a large class of groups which arises as subgroups of Thompson group F and reveal the relation between the skein theory of the subfactor planar algebra and the presentations of subgroups related to the corresponding unitary representation. Specifically, we answer a question by Jones about the 3-colorable subgroup.

Some isomorphism results for Thompson-like groups V n (G)

We find some perhaps surprising isomorphism results for the groups {V n (G)}, where V n (G) is a supergroup of the Higman–Thompson group V n for n ∈ N and G ≤ S n , the symmetric group on n points. These groups, introduced by Farley and Hughes, are the groups generated by V n and the tree automorphisms [α] g defined as follows. For each g ∈ G and each node α in the infinite rooted n-ary tree, the automorphisms [α] g acts iteratively as g on the child leaves of α and every descendent of α. In particular, we show that V n ≅ V n (G) if and only if G is semiregular (acts freely on n points), as well as some additional sufficient conditions for isomorphisms between other members of this family of groups. Essential tools in the above work are a study of the dynamics of the action of elements of V n (G) on the Cantor space, Rubin’s Theorem, and transducers from Grigorchuk, Nekrashevych, and Suschanskiĭ’s rational group on the n-ary alphabet.

On subgroups of R. Thompson’s group $F$

We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones' subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings' core graphs, and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>...$ such that $\cap H_i=\{1\}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.

Full groups of Cuntz–Krieger algebras and Higman–Thompson groups

In this paper, we will study representations of the continuous full group \Gamma_A of a one-sided topological Markov shift (X_A,\sigma_A) for an irreducible matrix A with entries in \{0,1\} as a generalization of Higman–Thompson groups V_N, 1 &lt; N \in {\mathbb{N}} . We will show that the group \Gamma_A can be represented as a group \Gamma_A^{\operatorname{tab}} of matrices, called A -adic tables, with entries in admissible words of the shift space X_A , and a group \Gamma_A^{\operatorname{PL}} of right continuous piecewise linear functions, called A -adic PL functions, on [0,1] with finite singularities.

Tarski monoids: Matui’s spatial realization theorem

This paper continues the study of a class of inverse monoids, called Tarski monoids, that can be regarded as non-commutative generalizations of the unique countable, atomless Boolean algebra. These inverse monoids are related to a class of étale topological groupoids under a non-commutative generalization of classical Stone duality and, significantly, they arise naturally in the theory of dynamical systems as developed by Matui. We are thereby able to reinterpret a theorem of Matui (à la Rubin) on a class of étale groupoids as an equivalent theorem about a class of Tarski monoids: two simple Tarski monoids are isomorphic if and only if their groups of units are isomorphic. The inverse monoids in question may also be viewed as countably infinite generalizations of finite symmetric inverse monoids. Their groups of units therefore generalize the finite symmetric groups and include amongst their number the Thompson groups $$V_{n}$$ .

Non-inner amenability of the Thompson groups T and V

In this paper we prove that the Thompson groups T and V are not inner amenable. In particular, their group von Neumann algebras do not have property Γ. Moreover, we prove that if the reduced group C⁎-algebra of T is simple, then the Thompson group F is non-amenable. Furthermore, we give a few new equivalent characterizations of amenability of F.