Thompson's Group Research Articles

Twisted Brin–Thompson groups

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 a family of representations of the Higman–Thompson groups

Abstract We obtain an uncountable family of inequivalent and irreducible representations of the Higman–Thompson groups F n ⊂ T n ⊂ V n F_{n}\subset T_{n}\subset V_{n} . This is accomplished by considering a family of representations of the Higman–Thompson groups V n V_{n} that arise from representations of Cuntz algebras, each one acting on a Hilbert space built upon the orbit of a point x ∈ [ 0 , 1 ) x\in[0,1) under the dynamical system Φ ⁢ ( x ) = n ⁢ x ( mod 1 ) \Phi(x)=nx\pmod{1} . Every such representation is retrieved through the action of V n V_{n} on orb ⁡ ( x ) \operatorname{orb}(x) , and their restrictions to the subgroups F n F_{n} and T n T_{n} of V n V_{n} are studied using properties of the groups.

Arborescence of positive Thompson links

We show that the links associated with positive elements of the Thompson group $F$ coincide with the closures of bipartite arborescent tangles.

Irrational-slope versions of thompson's groups T and V

AbstractIn this paper, we consider the $T$- and $V$-versions, ${T_\tau }$ and ${V_\tau }$, of the irrational slope Thompson group ${F_\tau }$ considered in J. Burillo, B. Nucinkis and L. Reeves [An irrational-slope Thompson's group, Publ. Mat. 65 (2021), 809–839]. We give infinite presentations for these groups and show how they can be represented by tree-pair diagrams similar to those for $T$ and $V$. We also show that ${T_\tau }$ and ${V_\tau }$ have index-$2$ normal subgroups, unlike their original Thompson counterparts $T$ and $V$. These index-$2$ subgroups are shown to be simple.

On the spread of infinite groups

AbstractA group is $\frac 32$-generated if every non-trivial element is part of a generating pair. In 2019, Donoven and Harper showed that many Thompson groups are $\frac 32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac 32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups $G_1,\, G_2,\, \ldots$ of the Houghton group $\operatorname {FSym}(\mathbb {Z})\rtimes \mathbb {Z}$. We then show that $G_1$ and $G_2$ are $\frac 32$-generated and investigate the related notion of spread for these groups. We are able to show that they have finite spread at least 2. These are, therefore, the first infinite groups to be shown to have finite positive spread, and the first to be shown to have spread at least 2 (other than $\mathbb {Z}$ and the Tarski monsters, which have infinite spread). As a consequence, for each $k\in \{2,\, 3,\, \ldots \}$, we also have that $G_{2k}$ is index $k$ in $G_2$ but $G_2$ is $\frac 32$-generated whereas $G_{2k}$ is not.

Direct decompositions of groups of piecewise linear homeomorphisms of the unit interval

We study subgroups of the group [Formula: see text] of piecewise linear orientation-preserving homeomorphisms of the unit interval [Formula: see text] that are differentiable everywhere except at finitely many real numbers, under the operation of composition. We provide a criterion for any two subgroups of [Formula: see text] which are direct products of finitely many indecomposable non-commutative groups to be non-isomorphic. As its application, we give a necessary and sufficient condition for any two subgroups of the R. Thompson group [Formula: see text] that are stabilizers of finite sets of numbers in the interval [Formula: see text] to be isomorphic, thus solving a problem by G. Golan and M. Sapir. We also show that if two stabilizers are isomorphic, then they are conjugate inside a certain group [Formula: see text].

R. Thompson's group $F$ and the amenability problem

В центре внимания данной работы находится группа Ричарда Томпсона $F$, открытая в 60-х годах прошлого века. Ей посвящено большое число работ. Нас интересует в первую очередь знаменитая проблема аменабельности этой группы, поставленная Россом Гейганом в 1979 г. Предпринимались многочисленные попытки решить еe как в том, так и в другом направлении, но она остаeтся открытой. В данном обзоре мы излагаем наиболее важные известные свойства этой группы, касающиеся проблемы слов, представления элементов группы кусочно линейными функциями, а также диаграммами и другими геометрическими объектами. Излагаются классические результаты М. Брина и К. Сквайера, касающиеся свободных подгрупп и тождеств. Мы включили изложение более современных важных результатов, относящихся к свойствам графов Кэли (конструкция Белка-Брауна), а также теорему Бартольди о свойствах уравнений в групповых кольцах. Отдельно рассматриваются критерии (не)аменабельности групп, полезные для работы по основной проблеме. В конце мы приводим ряд своих результатов о структуре графов Кэли и новом алгоритме решения проблемы слов. Библиография: 69 названий.

Some embeddings between symmetric R. thompson groups

Let $m\leqslant n\in \mathbb {N}$, and $G\leqslant \operatorname {Sym}(m)$ and $H\leqslant \operatorname {Sym}(n)$. In this article, we find conditions enabling embeddings between the symmetric R. Thompson groups ${V_m(G)}$ and ${V_n(H)}$. When $n\equiv 1 \mod (m-1)$, and under some other technical conditions, we find an embedding of ${V_n(H)}$ into ${V_m(G)}$ via topological conjugation. With the same modular condition, we also generalize a purely algebraic construction of Birget from 2019 to find a group $H\leqslant \operatorname {Sym}(n)$ and an embedding of ${V_m(G)}$ into ${V_n(H)}$.

Boundary maps, germs and quasi-regular representations

We investigate the tracial and ideal structures of C⁎-algebras of quasi-regular representations of stabilizers of boundary actions.Our main tool is the notion of boundary maps, namely Γ-equivariant unital completely positive maps from Γ-C⁎-algebras to C(∂FΓ), where ∂FΓ denotes the Furstenberg boundary of a group Γ.For a unitary representation π coming from the groupoid of germs of a boundary action, we show that there is a unique boundary map on Cπ⁎(Γ). Consequently, we completely describe the tracial structure of the C⁎-algebras Cπ⁎(Γ), and for any Γ-boundary X, we completely characterize the simplicity of the C⁎-algebras generated by the quasi-regular representations λΓ/Γx associated to stabilizer subgroups Γx for any x∈X.As an application, we show that the C⁎-algebra generated by the quasi-regular representation λT/F associated to Thompson's groups F≤T does not admit traces and is simple.

Random generation of Thompson group F

We prove that under two natural probabilistic models (studied by Cleary, Elder, Rechnitzer and Taback), the probability of a random pair of elements of Thompson group F generating the entire group is positive. We also prove that for any k-generated subgroup H of F which contains a “natural” copy of F, the probability of a random (k+2)-generated subgroup of F coinciding with H is positive.

Geometric structures related to the braided Thompson groups

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $${{\,\mathrm{F}\,}}_\infty $$ . The proof utilized certain contractible cube complexes, which in this paper we prove are $${{\,\mathrm{CAT}\,}}(0)$$ . We then use this fact to compute the geometric invariants $$\Sigma ^m(F_{{\text {br}}})$$ of the pure braided Thompson group $$F_{{\text {br}}}$$ . Only the first invariant $$\Sigma ^1(F_{{\text {br}}})$$ was previously known. A consequence of our computation is that as soon as a subgroup of $$F_{{\text {br}}}$$ containing the commutator subgroup $$[F_{{\text {br}}},F_{{\text {br}}}]$$ is finitely presented, it is automatically of type $${{\,\mathrm{F}\,}}_\infty $$ .

Group action with finite orbits on local dendrites

It is shown that the restriction of the action of any group with finite orbit on the minimal sets of local dendrites is equicontinuous. Consequently, we obtain that the action of any amenable group and Thompson group restricted to any minimal sets of dendrite is equicontinuous. We further provide a class of non-amenable groups whose action on the minimal sets of local dendrites is equicontinuous. Moreover, we extend some of our results to dendron. We further give a characterization of the set of invariant probability measures and its extreme points.

Normalish Amenable Subgroups of the R. Thompson Groups

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.

On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees

Consider an infinite homogeneous tree \mathcal{T}_n of valence n+1 , its group Aut (\mathcal{T}_n) of automorphisms, and the group Hier (\mathcal{T}_n) of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of \mathcal{T}_n that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut (\mathcal{T}_n) is spherical in Hier (\mathcal{T}_n) , i.e., any irreducible unitary representation of Hier (\mathcal{T}_n) contains at most one Aut (\mathcal{T}_n) -fixed vector. We present a combinatorial description of the space of double cosets of Hier (\mathcal{T}_n) with respect to Aut (\mathcal{T}_n) and construct a “new” family of spherical representations of Hier (\mathcal{T}_n) . We also show that the Thompson group Th has PSL (2,\mathbb{Z}) -spherical unitary representations.

Erratum to The braided Thompson's groups are of type F∞ (J. reine angew. Math. 718 (2016), 59–101)

Article Erratum to The braided Thompson's groups are of type F∞ (J. reine angew. Math. 718 (2016), 59–101) was published on September 1, 2021 in the journal Journal für die reine und angewandte Mathematik (volume 2021, issue 778).

Cayley polynomial–time computable groups

We propose a new generalization of Cayley automatic groups, varying the time complexity of computing multiplication, and language complexity of the normal form representatives. We first consider groups which have normal form language in the class C and multiplication by generators computable in linear time on a certain restricted Turing machine model (position–faithful one–tape). We show that many of the algorithmic properties of automatic groups are preserved, prove various closure properties, and show that the class is quite large. We then generalize to groups which have normal form language in the class C and multiplication by generators computable in polynomial time on a (standard) Turing machine. Of particular interest is when C=REG (the class of regular languages). We prove that REG–Cayley polynomial–time computable groups include all finitely generated nilpotent groups, the wreath product Z2≀Z2, and Thompson's group F.

Convergence Towards the End Space for Random Walks on Schreier Graphs

We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ defined by this action for some fixed generating set. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the induced random walk is transient, it converges towards the space of ends of $\Gamma$. As a corollary we obtain that for a probability measure with a finite first moment on Thompson's group $F$, the support of which generates $F$ as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary. Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng.

Complexity among the finitely generated subgroups of Thompson's group

We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group F which is strictly well-ordered by the embeddability relation of type \epsilon_0 +1 . All except the maximum element of this family (which is F itself) are elementary amenable groups. In fact we also obtain, for each \alpha < \epsilon_0 , a finitely generated elementary amenable subgroup of F whose EA-class is \alpha + 2 . These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with \mathbf Z + \mathbf Z , \mathbf Z \wr \mathbf Z , and the Brin–Navas group B . We also give an example of a pair of finitely generated elementary amenable subgroups of F with the property that neither is embeddable into the other.

Confined subgroups of Thompson's group $F$ and its embeddings into wobbling groups

We obtain a characterisation of confined subgroups of Thompson's group F . As a result, we deduce that the orbital graph of a point under an action of F has uniformly subexponential growth if and only if this point is fixed by the commutator subgroup. This allows us to prove non-embeddability of F into wobbling groups of graphs with uniformly subexponential growth.

On the oriented Thompson subgroup F→3 and its relatives in higher Brown–Thompson groups

A few years ago the so-called oriented subgroup [Formula: see text] of the Thompson group [Formula: see text] was introduced by V. Jones while investigating the connections between subfactors and conformal field theories. In the coding of links and knots by elements of [Formula: see text] it corresponds exactly to the oriented ones. Thanks to the work of Golan and Sapir, [Formula: see text] provided the first example of a maximal subgroup of infinite index in [Formula: see text] different from the parabolic subgroups that fix a point in [Formula: see text]. In this paper we investigate possible analogues of [Formula: see text] in higher Thompson groups [Formula: see text], with [Formula: see text], introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup [Formula: see text] of [Formula: see text], as described recently by Jones, and prove in particular that it gives rise to a non-parabolic maximal subgroup of infinite index in [Formula: see text] and that the corresponding quasi-regular representation is irreducible.