On identities in Thompson's group

Circuits, coNP-completeness, and the groups of Richard Thompson

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 G_{3,1}.

Presentations of higher dimensional Thompson groups

In a previous paper, we defined a higher dimensional analog of Thompson's group V, and proved that it is simple, infinite, finitely generated, and not isomorphic to any of the known Thompson groups. There are other Thompson groups that are infinite, simple and finitely presented. Here we show that the new group is also finitely presented by calculating an explicit finite presentation.

Quasi-isometrically embedded subgroups of Thompson's group F

The goal of this paper is to construct quasi-isometrically embedded subgroups of Thompson's group $F$ which are isomorphic to $\fz^n$ for all $n$. A result estimating the norm of an element of Thompson's group is found. As a corollary, Thompson's group is seen to be an example of a finitely presented group which has an infinite-dimensional asymptotic cone.

The Dehn Function of Richard Thompson's Group F is Quadratic

We prove that the Dehn function (that is, the smallest isoperimetric function) of the Richard Thompson's group F is quadratic.

仙人掌汤普森群的表示

Presentations of the cactus Thompson group

本研究给出由Witzel和Zaremsky引入的仙人掌汤普森群（记为$V_{\mathrm{mock}}$）的无限与有限表示，该群是汤普森群与仙人掌群结合的产物。

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

In this paper, we construct embeddings of right-angled Artin groups into higher dimensional Thompson groups. In particular, we embed every right-angled Artin groups into n-dimensional Thompson group, where n is the number of complementary edges in the defining graph.

Thompson群T具有二次Dehn函数

Thompson's group T has quadratic Dehn function

本文证明了Thompson群T及所有Higman-Thompson群T_n的Dehn函数为二次的，这是对其几何性质的重要刻画。

Brin-Thompson 群的子群上的若干障碍及部分扭曲 Brin-Thompson 群

Some obstructions on the subgroups of the Brin-Thompson groups and a selection of twisted Brin-Thompson groups

受 Thompson 群 V 子群障碍研究启发，本文探索了高维与无限维 Brin-Thompson 群及其扭曲群，发现它们在挠子群上具有类似障碍，并排除了部分 Baumslag-Solitar 群作为其子群。

A note on stable commutator length in braided Ptolemy-Thompson groups

In this note, we show that the sets of all stable commutator lengths in the braided Ptolemy-Thompson groups are equal to non-negative rational numbers.

关于汤普森群V度量的注记

A Note on the Metric of Thompson's group V

本文改进了Birget于2004年对汤普森群V字度量的界，结果与已知的F和T群的界一致。

On the Baker's map and the Simplicity of the Higher Dimensional Thompson Groups nV

We show that the baker's map is a product of transpositions (particularly pleasant involutions), and conclude from this that an existing very short proof of the simplicity of Thompson's group V applies with equal brevity to the higher dimensional Thompson groups nV.

Metric properties of the braided Thompson's groups

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 $B_{\infty}$ 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.

Cayley graphs of R. Thompson's group F: new estimates for the density

We improve some known estimates for the density of the Cayley graph of Thompson's group F in standard generators.

Braided Brin-Thompson Groups

We construct braided versions $sV_{br}$ of the Brin-Thompson groups $sV$ and prove that they are of type $F_\infty$. The proof involves showing that the matching complexes of colored arcs on surfaces are highly connected.

Invariable generation of Thompson groups

A subset $S$ of a group $G$ invariably generates $G$ if $G= \langle s^{g(s)} | s \in S\rangle$ for every choice of $g(s) \in G,s \in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this paper, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, whereas Thompson groups $T$ and $V$ are not invariable generated.

Fixed points of finite groups acting on generalised Thompson groups

We study centralisers of finite order automorphisms of generalisations of Thompson's group F and conjugacy classes of finite subgroups in finite extensions of these groups. In particular, we show that centralisers of finite automorphisms in these generalised Thompson groups either admit a finite type classifying space or are not finitely generated. As an application we deduce results about the Bredon type of such finite extensions.

Higher dimensional Thompson groups have Serre's property FA

The Thompson group V is a subgroup of the homeomorphism group of the Cantor set. Brin defined higher dimensional Thompson groups nV as generalizations of V. We prove that nV has Serre's property FA, for every n. This is a generalization of the corresponding result of Farley, who studied V.

Fast growth in Folner sets for Thompson's group F

The purpose of this note is to prove a lower bound on the Folner function for Thompson's groups F.

Thompson's group F is not strongly amenable

We show that Thompson's group $F$ has a topological action on a compact metric space that is proximal and has no fixed points.

Embedding Right-Angled Artin Groups into Brin-Thompson Groups

We prove that every finitely-generated right-angled Artin group can be embedded into some Brin-Thompson group $nV$. It follows that many other groups can be embedded into some $nV$ (e.g., any finite extension of any of Haglund and Wise's special groups), and that various decision problems involving subgroups of $nV$ are unsolvable.

关于汤普森群F的FC-中心的注记

A Note on the FC-Center of Thompson's Group $F$

本研究证明汤普森群F的FC-中心（含有限共轭类元素的子群）是平凡的，为群论相关研究提供了重要结论。

Traveller Salesman Property and Richard Thompson's Group $F$

Recently Akhmedov introduced a property of finitely generated groups called the Traveller Salesman Property (TSP). It was shown by Thurston that TSP implies non-amenability. Akhmedov conjectured that R. Thompson's group $F$ has TSP. The aim of this article is to disprove this conjecture.

New embeddings between the Higman-Thompson groups

We give a direct proof that all Higman-Thompson groups of the form $G_{k,1}$ (for $k \ge 2$) are embedded in one another, which is a recent result of N. Matte Bon. This extends the embeddings given by Higman in 1974.

Cohomological finiteness properties of the Brin-Thompson-Higman groups 2V and 3V

We show that Brin's generalisations $2V$ and $3V$ of the Thompson-Higman group $V$ are of type $FP_\infty$. Our methods also give a new proof that both groups are finitely presented.

Some Remarks on the Braided Thompson Group BV

Matthew Brin and Patrick Dehornoy independently discovered a braided version BV of Thompson's group V. In this paper, we discuss some properties of BV that might make the group interesting for group based cryptography. In particular, we show that BV does not admit a non-trivial linear representation.

Limits of Thompson's group F

Let F be the Thompson's group. We study the structure of F-limit groups. Consider a sequence of groups marked by three elements, each isomorphic to F. Assume that the this sequence is convergent in the space of marked groups. We prove that no HNN-extension of F over a cyclic subgroup occurs as an F-limit group of such a sequence.

The infinite associahedron and R.J. Thompson's group T

In this paper we construct a cellular complex which is an infinite analogue to Stasheff's associahedra. We prove that it is contractible and state that its (combinatorial) automorphism group is isomorphic to a semi-direct product of R.J. Thompson's group $T$ with $\mathbb{Z}/2\mathbb{Z}$.

The Free Group of Rank 2 is a Limit of Thompson's Group F

We show that the free group of rank 2 is a limit of 2-markings of Thompson's group F in the space of all 2-marked groups. More specifically, we find a sequence of generating pairs for F so that as one goes out the sequence, the length of the shortest relation satisfied by the generating pair goes to infinity.

高维Thompson群的发散函数

Divergence functions of higher-dimensional Thompson's groups

本文证明高维Thompson群具有线性发散函数，并由此推出其渐近锥均无割点。

纯无限辫子Thompson群

Pure infinitely braided Thompson groups

该研究证明无限辫子Thompson群的纯子群具有双可序性，并为所有有限生成纯子群提供了显式生成元集，推进了群论相关研究。

Embeddings into Thompson's groups from quasi-median geometry

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 embeds 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 graph structure of graph groups that are subgroups of Thompson's group $V$

We determine exactly which graph products, also known as Right Angled Artin Groups, embed into Richard Thompson's group $V$. It was shown by Bleak and Salazar-Diaz that $\mathbb{Z}^2 * \mathbb{Z}$ was an obstruction. We show that this is the only obstruction. This is shown by proving a graph theory result giving an alternate description of simple graphs without an appropriate induced subgraph.

Presentations for the higher dimensional Thompson's groups nV

In his papers [2], [3] Brin introduced the higher dimensional Thompson groups nV which are generalizations to the Thompson's group V of self-homeomorphisms of the Cantor set and found a finite set of generators and relations in the case n = 2. We show how to generalize his construction to obtain a finite presentation for every positive integer n. As a corollary, we obtain another proof that the groups nV are simple (first proved by Brin in [4]).

Pure braid subgroups of braided Thompson's groups

We describe pure braided versions of Thompson's group F. These groups, $BF$ and $\hat{BF}$, are subgroups of the braided versions of Thompson's group V, introduced by Brin and Dehornoy. Unlike V, elements of F are order-preserving self-maps of the interval and we use pure braids together with elements of F thus preserving order. We define these groups and give normal forms for elements and describe infinite and finite presentations of these groups.

The Dehn function of Richard Thompson's group $T$

V.S.Guba had proved that the R.Thompson group $T$ satisfies polynomial isoperimetric inequality and $Φ_T(n)\preceq n^7$, where $Φ_T$ is the Dehn function of group $T$. In this paper, we show that $Φ_T(n)\preceq n^5$.

Obstructions for subgroups of Thompson's group $V$

Thompson's group $V$ has a rich variety of subgroups, containing 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 $V$.

Subgroups of R. Thompson's Group F that are Isomorphic to F

This paper studies when a pair of elements in F are the images of the standard generators of F under a self monomorphism.

Addendum to "Commensurations and Subgroups of Finite Index of Thompson's Group F"

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.

Monoids that map onto the Thompson-Higman groups

A slight modification of the definition of the Thompson-Higman groups G_k1 and F_k1 leads to inverse monoids that map onto G_k1 (respectively F_k1), and that have interesting properties: they are finitely generated, and residually finite. These inverse monoids are closely related to the suffix expansion of G_k1 (respectively F_k1).

Computational explorations in Thompson's group F

We describe the results of some computational explorations in Thompson's group F. We describe experiments to estimate the cogrowth of F with respect to its standard finite generating set, designed to address the subtle and difficult question whether or not Thompson's group is amenable. We also describe experiments to estimate the exponential growth rate of F and the rate of escape of symmetric random walks with respect to the standard generating set.

Conjugacy and Dynamics in Thompson's Groups

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.

On Thompson's conjecture for finite simple groups

Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group.

Higman-Thompson群与右角Coxeter群的紧致性质

Higman-Thompson groups and profinite properties of right-angled Coxeter groups

本文证明了右角Coxeter群在Coxeter群中具有profinite刚性，并展示了某些RACG具有无限profinite genus，同时推广了Higman关于Thompson群$V$的生成结果。

Superharmonic functions on the Lamplighter graph of Thompson's group F

The goal is to extend a non-standard amenability test for groups, based on random walks and superharmonic functions, to group actions on sets, and to apply it to Thompson's group F using certain properties of extensive amenability. While no conclusive answer regarding the amenability of F is given, the approach is helpful in developing a new potentially useful criterion and testing it on a significant subclass of superharmonic functions.

The conjugacy problem for Thompson-like groups

In this paper we generalize techniques of Belk-Matucci to solve the conjugacy problem for every Thompson-like group $V_n(H)$, where $n \geq 2$ and $H$ is a subgroup of the symmetric group on $n$ elements. We use this to prove that, if $n \neq m$, $V_n(H)$ is not isomorphic to $V_m(G)$ for any $H,G$.

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 fam…

Irrational-slope versions of Thompson's groups $T$ and $V$

In this paper we consider the $T$- and $V$- versions, $T_τ$ and $V_τ$ , of the irrational slope Thompson group $F_τ$ considered in [3]. 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 $T_τ$ and $V_τ$ have index-2 normal subgroups, unlike their original Thompson counterparts $T$ and $V$. These index-2 subgroups are shown to be simple.

On Thompson's conjecture for alternating and symmetric groups

For a finite group $G$ denote by $N(G)$ the set of conjugesy class sizes of $G$. We show that every finite group $G$ with the property $N(G)=N(Alt_n), n>4$ or $N(G)=N(Sym_n), n>22$ is non-solvable.

Thompson群V的Dehn函数至多为六次

The Dehn function of Thompson's group $V$ is at most sextic

本文将Thompson群V的Dehn函数上界从n¹¹改进为n⁶，同时通过显式van Kampen图探究其关系复杂性，补充了Thompson群Dehn函数的相关研究进展。

Topological full groups and t.d.l.c. completions of Thompson's $V$

We show how all topological full groups coming from a one-sided irreducible shift of finite type, as studied by Matui, can be re-interpreted as groups of colour-preserving tree almost automorphisms. As an application, we show that they admit t.d.l.c. completions of arbitrary finite local prime content. This applies in particular to Thompson's $V$.

Thompson群T的无$\boldsymbol{\mathbb{Z}^2}$子群是虚自由的

$\mathbb{Z}^2$-free subgroups of Thompson's $T$ are virtually free

本文证明Thompson群T的有限生成且不含任意高阶阿贝尔群的子群为虚阿贝尔或虚自由，进而得出曲面群不能嵌入T，解答了Belk与Moore的公开问题。

Conjugacy, roots, and centralizers in Thompson's group $F$

We complete the program begun by Brin and Squier of characterising conjugacy in Thompson's group $F$ using the standard action of $F$ as a group of piecewise linear homeomorphisms of the unit interval.

Thompson's conjecture for alternating group

Let $G$ be a finite group, and let $N(G)$ be the set of sizes of its conjugacy classes. We show that if a finite group $G$ has trivial center and $N(G)$ equals to $N(Alt_n)$ or $N(Sym_n)$ for $n\geq 23$, then $G$ has a composition factor isomorphic to an alternating group $Alt_k$ such that $k\leq n$ and the half-interval $(k, n]$ contains no primes. As a corollary, we prove the Thompson's conjecture for simple alternating groups.

The chameleon groups of Richard J. Thompson: automorphisms and dynamics

The automorphism groups of several of Thompson's countable groups of piecewise linear homeomorphisms of the line and circle are computed and it is shown that the outer automorphism groups of these groups are relatively small. These results can be interpreted as stability results for certain structures of PL functions on the circle. Machinery is developed to relate the structures on the circle to corresponding structures on the line.