Thompson's Group Research Articles

Decomposition of Thompson group representations arising from Cuntz algebras

Abstract For every integer n ≥ 2 n\geq 2 , we consider a family { π w } w ∈ { 0 , 1 , … , n − 1 } N \{\pi_{w}\}_{w\in\{0,1,\ldots,n-1\}^{\mathbb{N}}} of irreducible representations of the Cuntz algebra O n \mathcal{O}_{n} . All these representations (except one) are shown to be equivalent to those arising from the orbits of the interval map dynamical system ( I , f ) (I,f) with f ⁢ ( x ) = n ⁢ x ⁢ ( mod ⁢ 1 ) f(x)=nx\ (\mathrm{mod}\ 1) . We consider the embeddings V = V 2 ⊂ V n ⊂ O n V=V_{2}\subset V_{n}\subset\mathcal{O}_{n} of the Thompson group 𝑉 in the Higman–Thompson group V n V_{n} obtained by Birget and then from V n V_{n} into O n \mathcal{O}_{n} which was obtained independently by Birget and Nekrashevych. The restriction of π w \pi_{w} to V n V_{n} is still irreducible; however, the restriction of π w \pi_{w} to 𝑉 is no longer irreducible, and we obtain the corresponding irreducible decomposition in terms of quasi-regular representations.

The Borel complexity of the space of left‐orderings, low‐dimensional topology, and dynamics

AbstractWe develop new tools to analyze the complexity of the conjugacy equivalence relation , whenever is a left‐orderable group. Our methods are used to demonstrate nonsmoothness of for certain groups of dynamical origin, such as certain amalgams constructed from Thompson's group . We also initiate a systematic analysis of , where is a 3‐manifold. We prove that if is not prime, then is a universal countable Borel equivalence relation, and show that in certain cases the complexity of is bounded below by the complexity of the conjugacy equivalence relation arising from the fundamental group of each of the JSJ pieces of . We also prove that if is the complement of a nontrivial knot in then is not smooth, and show how determining smoothness of for all knot manifolds is related to the L‐space conjecture.

The $p$-colorable subgroup of Thompson’s group

V. F. R. Jones introduced a method of constructing knots and links from elements of Thompson’s group F by using its unitary representations. He also defined several subgroups of F as the stabilizer subgroups and some researchers studied them algebraically. One of the subgroups is called the 3 -colorable subgroup \mathcal{F} , and the authors proved that all knots and links obtained from non-trivial elements of \mathcal{F} are 3 -colorable. In this paper, for any odd integer p greater than two, we define the p -colorable subgroup of F whose non-trivial elements yield p -colorable knots and links and show it is isomorphic to a certain Brown–Thompson group.

Conway Rational Tangles and the Thompson Group

Conway Rational Tangles and the Thompson Group

Geometry of the Thompson group

We enhance the Thompson–Smith construction [34], [35], [32], [33] of the Thompson sporadic simple group E. As a result, (a) we obtain a conceptual explanation of the dichotomy between Lie and finite cases in terms of representations of an (L2(8):3)-subgroup; (b) along with 25⋅L5(2) and 2+1+8⋅A9, we construct in E subgroups D43(2):3, (F21×L3(2)):2, (G2(3)×3):2 and U3(8):6; (c) we consider the coset geometry of the above subgroups and identify the corresponding geometric presentation with that by Havas–Soicher–Wilson [14]; (d) this amounts to a new geometric construction and uniqueness proof for E; (e) we deliver a self-contained construction of the Dempwolff–Thompson orthogonal decomposition on the E8-Lie algebra, along with the stabiliser 25+10⋅L5(2) of this decomposition in the Lie group E8(C). The project was accomplished within a general approach to finite simple groups through their simply connected geometries [19]. The work was further stimulating by the recent reappearance of E in the Moonshine context [2], [13].

Braided Thompson groups with and without quasimorphisms

Braided Thompson groups with and without quasimorphisms

Pure infinitely braided Thompson groups

We show that pure subgroups of infinitely braided Thompson’s are bi-orderable. For every finitely generated pure subgroup, we give explicit sets of generators.

Finiteness properties of locally defined groups

Let X be a set and let S be an inverse semigroup of partial bijections of X . Thus, an element of S is a bijection between two subsets of X , and the set S is required to be closed under the operations of taking inverses and compositions of functions. We define \Gamma_{S} to be the set of self-bijections of X in which each \gamma \in \Gamma_{S} is expressible as a union of finitely many members of S . This set is a group with respect to composition. The groups \Gamma_{S} form a class containing numerous widely studied groups, such as Thompson’s group V , the Nekrashevych–Röver groups, Houghton’s groups, and the Brin–Thompson groups nV , among many others. We offer a unified construction of geometric models for \Gamma_{S} and a general framework for studying the finiteness properties of these groups.

Property $\mathrm{(NL)}$ for group actions on hyperbolic spaces (with an appendix by Alessandro Sisto)

We introduce Property \mathrm{(NL)} , which indicates that a group does not admit any (isometric) action on a hyperbolic space with loxodromic elements. In other words, such a group G can only admit elliptic or horocyclic hyperbolic actions, and consequently its poset of hyperbolic structures \mathcal{H}(G) is trivial. It turns out that many groups satisfy this property, and we initiate the formal study of this phenomenon. Of particular importance is the proof of a dynamical criterion in this paper that ensures that groups with “rich” actions on compact Hausdorff spaces have Property \mathrm{(NL)} . These include many Thompson-like groups, such as V, T and even twisted Brin–Thompson groups, which implies that every finitely generated group quasi-isometrically embeds into a finitely generated simple group with Property \mathrm{(NL)} . We also study the stability of the property under group operations and explore connections to other fixed point properties. In the appendix (by Alessandro Sisto), we describe a construction of cobounded actions on hyperbolic spaces starting from non-cobounded ones that preserves various properties of the initial action.

A class of rearrangement groups that are not invariably generated

AbstractA group is invariably generated if there exists a subset such that, for every choice for , the group is generated by . Gelander, Golan, and Juschenko (J. Algebra 478 (2016), 261–270) showed that Thompson groups and are not invariably generated. Here, we generalize this result to the larger setting of rearrangement groups, proving that any subgroup of a rearrangement group that has a certain transitive property is not invariably generated.

Cayley Linear-Time Computable Groups

This paper looks at the class of groups admitting normal forms for which the right multiplication by a group element is computed in linear time on a multi-tape Turing machine. We show that the groups $\mathbb{Z}_2 \wr \mathbb{Z}^2$, $\mathbb{Z}_2 \wr \mathbb{F}_2$ and Thompson's group $F$ have normal forms for which the right multiplication by a group element is computed in linear time on a $2$-tape Turing machine. This refines the results previously established by Elder and the authors that these groups are Cayley polynomial-time computable.

Divergence Property of the Brown-Thompson Groups and Braided Thompson Groups

AbstractGolan and Sapir proved that Thompson’s groups F, T and V have linear divergence. In the current paper, we focus on the divergence property of several generalisations of the Thompson groups. We first consider the Brown-Thompson groups $$F_n$$ F n , $$T_n$$ T n and $$V_n$$ V n (also called Brown-Higman-Thompson group in some other context) and find that these groups also have linear divergence functions. We then focus on the braided Thompson groups BF, $$\widehat{BF}$$ BF ^ and $$\widehat{BV}$$ BV ^ and prove that these groups have linear divergence. The case of BV has also been done independently by Kodama.

Weak commutativity, virtually nilpotent groups, and Dehn functions

The group \mathfrak{X}(G) is obtained from G\ast G by forcing each element g in the first free factor to commute with the copy of g in the second free factor. We make significant additions to the list of properties that the functor \mathfrak{X} is known to preserve. We also investigate the geometry and complexity of the word problem for \mathfrak{X}(G) . Subtle features of \mathfrak{X}(G) are encoded in a normal abelian subgroup W<\mathfrak{X}(G) that is a module over \mathbb{Z} Q , where Q= H_1(G,\mathbb{Z}) . We establish a structural result for this module and illustrate its utility by proving that \mathfrak{X} preserves virtual nilpotence, the Engel condition, and growth type – polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for \mathfrak{X}(G) when G lies in a class that includes Thompson's group F and all non-fibred Kähler groups. The word problem is soluble in \mathfrak{X}(G) if and only if it is soluble in G . The Dehn function of \mathfrak{X}(G) is bounded below by a cubic polynomial if G maps onto a non-abelian free group.

The Generation Problem in Thompson Group 𝐹

We show that the generation problem in Thompson’s group F F is decidable, i.e., there is an algorithm which decides if a finite set of elements of F F generates the whole F F . The algorithm makes use of the Stallings 2 2 -core of subgroups of F F , which can be defined in an analogous way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings 2 2 -core of subgroups of F F provides a solution to another algorithmic problem in F F . Namely, given a finitely generated subgroup H H of F F , it is decidable if H H acts transitively on the set of finite dyadic fractions D \mathcal D . Other applications of the study include the construction of new maximal subgroups of F F of infinite index, among which, a maximal subgroup of infinite index which acts transitively on the set D \mathcal D and the construction of an elementary amenable subgroup of F F which is maximal in a normal subgroup of F F .

Distortion element in the automorphism group of a full shift

Abstract We show that there is a distortion element in a finitely generated subgroup G of the automorphism group of the full shift, namely an element of infinite order whose word norm grows polylogarithmically. As a corollary, we obtain a lower bound on the entropy dimension of any subshift containing a copy of G, and that a sofic shift’s automorphism group contains a distortion element if and only if the sofic shift is uncountable. We obtain also that groups of Turing machines and the higher-dimensional Brin–Thompson groups $mV$ admit distortion elements; in particular, $2V$ (unlike V) does not admit a proper action on a CAT $(0)$ cube complex. In each case, the distortion element roughly corresponds to the SMART machine of Cassaigne, Ollinger, and Torres-Avilés [A small minimal aperiodic reversible Turing machine. J. Comput. System Sci.84 (2017), 288–301].

Amenability problem for Thompson's group $F$: state of the art

This is a survey of our recent results on the amenability problem for Thompson's group $F$. They mostly concern esimating the density of finite subgraphs in Cayley graphs of $F$ for various systems of generators, and also equations in the group ring of $F$. We also discuss possible approaches to solve the problem in both directions.

Ulam stability of lamplighters and Thompson groups

We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten p-norms. These include lamplighters Γ≀Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma \\wr \\Lambda $$\\end{document} where Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups F,F′,T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F, F', T$$\\end{document} and V. We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability. Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability. We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.

Finiteness properties for relatives of braided Higman–Thompson groups

We study the finiteness properties of the braided Higman–Thompson group bV_{d,r}(H) with labels in H\leq B_d , and bF_{d,r}(H) and bT_{d,r}(H) with labels in H\leq PB_d , where B_d is the braid group with d strings and PB_d is its pure braid subgroup. We show that for all d\geq 2 and r\geq 1 , the group bV_{d,r}(H) (resp. bT_{d,r}(H) or bF_{d,r}(H) ) is of type F_n if and only if H is. Our result in particular confirms a recent conjecture of Aroca and Cumplido.

Semi-simple actions of the Higman-Thompson groups T_n on finite-dimensional CAT(0) spaces

In this paper, we study isometric actions on finite-dimensional CAT(0) spaces for the Higman–Thompson groups T_n, which are generalizations of Thompson’s group T. It is known that every semi-simple action of T on a complete CAT(0) space of finite covering dimension has a global fixed point. After this result, we show that every semi-simple action of T_n on a complete CAT(0) space of finite covering dimension has a global fixed point. In the proof, we regard T_n as ring groups of homeomorphisms of S^1 introduced by Kim, Koberda and Lodha, and use general facts on these groups.

The 3-colorable subgroup of Thompson's group and tricolorability of links

Starting from the work by Jones on representations of Thompson's group F, subgroups of F with interesting properties have been defined and studied. One of these subgroups is called the 3-colorable subgroup F, which consists of elements whose “regions” given by their tree diagrams are 3-colorable. On the other hand, in his work on representations, Jones also gave a method to construct knots and links from elements of F. Therefore it is a natural question to explore a relationship between elements in F and 3-colorable links in the sense of knot theory. In this paper, we show that all elements in F give 3-colorable links.