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

Twisted conjugacy in Richard Thompson's group T

Let $f$ be an automorphism of a group $G$. Two elements $x, y$ in $G$ are said to be in the same $f$-twisted conjugacy class if there exists an element $z$ in $G$ such that $y=z x f(z^{-1})$. This is an equivalence relation known as $f$-twisted conjugacy. If the number $R(f)$ of $f$-twisted conjugacy classes is infinite for every automorphism $f$ of $G$ one says that $G$ has the $R_\infty$-property. We show that the Richard Thompson group $T$ has the $R_\infty$-property.

在 Richard Thompson 群景观中的有限表示单左序群

Finitely presented simple left-orderable groups in the landscape of Richard Thompson's groups

本文首次构造了实数线上保向同胚的有限表示单群，该类群具有类型 $F_{\infty}$、无限几何维数及非平凡齐次拟态，从而具有无限换位子宽度。

The homology of Richard Thompson's group F

In the early 1980s Ross Geoghegan and I calculated the homology of Richard Thompson's group F as a graded abelian group. It turns out that the homology admits a natural ring structure, which I calculate in this paper. As a byproduct, I obtain the ring structure on the cohomology.

The infinite simple group V of Richard J. Thompson: presentations by permutations

We show that one can naturally describe elements of R. Thompson's finitely presented infinite simple group $V$, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions. This perspective provides an intuitive explanation towards the simplicity of $V$ and also perhaps indicates a reason as to why it was one of the first discovered infinite finitely presented simple groups: it is (in some basic sense) a relative of the finite alternating groups. We find a natural infinite presentation for $V$ as a group generate…

The groups of Richard Thompson and complexity

We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We define the groups as partial transformation groups and we give a faithful representation in the Cuntz C*-algebra. For the finitely presented simple group T_fin (also known as V) we show that the word-length and the table size satisfy an n log n relation, just like the symmetric groups. We show that the word problem of T_fin belongs to the parallel complexity class AC^1 (a subclass of plynomial time). We show that the generalized word problem of T_fin is undecidable. We study the dis…

Thompson's Group F is Not Kahler

The purpose of this note is to prove that Richard Thompson's group F and variants of it studied by Ken Brown are not Kahler groups.

The further chameleon groups of Richard Thompson and Graham Higman: Automorphisms via dynamics for the Higman groups $G_{n,r}$

We describe, through the use of Rubin's theorem, the automorphism groups of the Higman-Thompson groups $G_{n,r}$ as groups of specific homeomorphisms of Cantor spaces $\mathfrak{C}_{n,r}$. This continues a thread of research begun by Brin, and extended later by Brin and Guzmán: to characterise the automorphism groups of the `Chameleon groups of Richard Thompson,' as Brin referred to them in 1996. The work here completes the first stage of that twenty-year-old program, containing (amongst other things) a characterisation of the automorphism group of $V$, which was the `last chameleon.' The home…

Monoid generalizations of the Richard Thompson groups

The groups G_{k,1} of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call M_{k,1}, and to inverse monoids, called Inv_{k,1}; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids M_{k,1} have connections with circuit complexity (studied in another paper). Here we prove that M_{k,1} and Inv_{k,1} are congruence-simple for all k. Their Green relations J and D are characterized: M_{k,1} and Inv_{k,1} are J-0-simple, and they have k-1 non-zero D-classes. They are submonoids of the multiplicative p…

The R- and L-orders of the Thompson-Higman monoid M_{k,1} and their complexity

We study the monoid generalization M_{k,1} of the Thompson-Higman groups, and we characterize the R- and the L-preorder of M_{k,1}. Although M_{k,1} has only one non-zero J-class and k-1 non-zero D-classes, the R- and the L-preorder are complicated; in particular, <_R is dense (even within an L-class), and <_L is dense (even within an R-class). We study the computational complexity of the R- and the L-preorder. When inputs are given by words over a finite generating set of M_{k,1}, the R- and the L-preorder decision problems are in P. The main result of the paper is that over a "circuit-like" …

A monoid version of the Brin-Higman-Thompson groups

We generalize the Brin-Higman-Thompson groups $n G_{k,1}$ to monoids $n M_{k,1}$, for $n \ge 1$ and $k \ge 2$, by replacing bijections by partial functions. The monoid $n M_{k,1}$ has $n G_{k,1}$ as its group of units, and is congruence-simple. Moreover, $n M_{k,1}$ is finitely generated, and for $n \ge 2$ its word problem is {\sf coNP}-complete. We also present new results about higher-dimensional joinless codes.

On the Properties of the Cayley Graph of Richard Thompson's Group F

We study some properties of the Cayley graph of the R.Thompson's group F in generators $x_0$, $x_1$. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant $K>0$, each vertex m…

The word problem of the Brin-Higman-Thompson groups

We show that the word problem of the Brin-Higman-Thompson group $n G_{k,1}$ is {\sf coNP}-complete for all $n \ge 2$ and all $k \ge 2$. For this we prove that $n G_{k,1}$ is finitely generated, and that $n G_{k,1}$ contains a subgroup of $2 G_{2,1}$ that can represent bijective circuits. We also show that for all $n \ge 1$ and $k \ge 2$: \ If $\,K = 1 + (k-1)\,N\,$ for some $N \ge 1$, then $n G_{K,1} \le n G_{k,1}$. In particular, $n G_{K,1} \le n G_{2,1}$ for all $K \ge 2$.

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.

Thompson's Group F is not Minimally Almost Convex

We prove that Richard Thompson's group F is not minimally almost convex with respect to the two standard generators. This improves upon a recent result of S. Cleary and J. Taback. We make use of the forest diagrams for elements of F introduced by J. Belk and K. Brown. These diagrams seem particularly well-suited for understanding the Cayley graph for the two standard generators.

Growth of positive words and lower bounds of the growth rate for Thompson's groups $F(p)$

Let $F(p)$, $p\ge2$ be the family of generalized Thompson's groups. Here F(2) is the famous Richard Thompson's group usually denoted by $F$. We find the growth rate of the monoid of positive words in $F(p)$ and show that it does not exceed $p+1/2$. Also we describe new normal forms for elements of $F(p)$ and, using these forms, we find a lower bound for the growth rate of $F(p)$ in its natural generators. This lower bound asymptotically equals $(p-1/2)\log_2 e+1/2$ for large values of $p$.

Twisted conjugacy in generalized Thompson groups of type F

If $ϕ$ is an automorphism of a group $G$ and $x,y\in G$, we say that $x$ and $y$ are $ϕ$-twisted conjugates if there exists an $z\in G$ such that $y=z.x.ϕ(z^{-1})$. This is an equivalence relation. If there are infinitely many $ϕ$-twisted conjugacy classes for every automorphism $ϕ$ of $G$ we say that $G$ has the $R_\infty$-property. We prove that the generalized Richard Thompson groups $F_n$ and $F(l,A,P)$ have the $R_\infty$-property.

Forest-skein groups I: between Vaughan Jones' subfactors and Richard Thompson's groups

Vaughan Jones discovered unexpected connections between Richard Thompson's group and subfactor theory while attempting to construct conformal field theories (in short CFT). Among other this founded Jones' technology: a powerful new method for constructing actions of fraction groups which had numerous applications in mathematical physics, operator algebras, group theory and more surprisingly in knot theory and noncommutative probability theory. We propose and outline a program in the vein of Jones' work but where the Thompson group is replaced by a family of groups that we name forest-skein gro…

Factorizations of the Thompson-Higman groups, and circuit complexity

We consider the subgroup lpG_{k,1} of length preserving elements of the Thompson-Higman group G_{k,1} and we show that all elements of G_{k,1} have a unique lpG_{k,1}.F_{k,1} factorization. This applies to the Thompson-Higman group T_{k,1} as well. We show that lpG_{k,1} is a ``diagonal'' direct limit of finite symmetric groups, and that lpT_{k,1} is a k^infinity Pr"ufer group. We find an infinite generating set of lpG_{k,1} which is related to reversible boolean circuits. We further investigate connections between the Thompson-Higman groups, circuits, and complexity. We show that elements of …

The Thompson-Higman monoids M_{k,i}: the J-order, the D-relation, and their complexity

The Thompson-Higman groups G_{k,i} have a natural generalization to monoids M_{k,i}, and inverse monoids Inv_{k,i}. We study some structural features of M_{k,i} and Inv_{k,i} and investigate the computational complexity of decision problems. The main interest of these monoids is their close connection with circuits and circuit complexity. The maximal subgroups of M_{k,1} are isomorphic to the groups G_{k,j} (1 \leq j \leq k-1); so we rediscover all the Thompson-Higman groups within M_{k,1}. The Green relations \leq_J and \equiv_D of M_{k,1} can be decided in deterministic polynomial time when …

On Jones' connections between subfactors, conformal field theory, Thompson's groups and knots

Surprisingly Richard Thompson's groups have recently appeared in Jones' subfactor theory. Vaughan Jones is famous for linking theories that are a priori completely disconnected; for instance, his celebrated polynomial for links emanating from subfactor theory. This note is about a new beautiful story in mathematics which results from a fortunate accident in the land of quantum field theory.

Jones' representations of R. Thompson's groups not induced by finite-dimensional ones

Given any linear isometry from a Hilbert space to its square one can explicitly construct a so-called Pythagorean unitary representation of Richard Thompson's group F. We introduce a condition on the isometry implying that the associated representation does not contain any induced representations by finite-dimensional ones. This provides the first result of this kind. We illustrate this theorem via a family of representations parametrised by the real 3-sphere for which all of them have this property except two sub-circles.

R. Thompson 群的毕达哥拉斯表示的分解

Decomposition of Pythagorean representations of R. Thompson's groups

针对由单个等距映射构造的 Thompson 群 $F,T,V$ 毕达哥拉斯酉表示，本文利用图技术将其分解为连续部分与原子部分，并完全刻画原子部分的单项表示来源。

Ideal structure of the C*-algebra of Thompson group T

In a recent paper Uffe Haagerup and Kristian Knudsen Olesen show that for Richard Thompson's group $T$, if there exists a finite set $H$ which can be decomposed as disjoint union of sets $H_1$ and $H_2$ with $\sum_{g\in H_1}π(g)=\sum_{h\in H_2}π(h)$ and such that the closed ideal generated by $\sum_{g\in H_1}λ(g)-\sum_{h\in H_2}λ(h)$ coincides with $C^*_λ(T)$, then the Richard Thompson group $F$ is not amenable. In particular, if $C_λ^*(T)$ is simple then $F$ is not amenable. Here we prove the converse, namely, if $F$ is not amenable then we can find two sets $H_1$ and $H_2$ with the above pro…

Interpretation of the Arithmetic in certain groups of piecewise affine permutations of an interval

The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group $F$ and some of its generalizations interpret the Arithmetic without parameters.

Interprétation de l'Arithmétique dans certains groupes de permutations affines par morceaux d'un intervalle

The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group $F$ and some of its generalizations interpret the Arithmetic without parameters.

Cuntz代数与R. Thompson群的表示的张量积

A tensor product for representations of the Cuntz algebra and of the R. Thompson groups

本文通过Pythagorean酉表示将Richard Thompson群$F,T,V$与Cuntz代数$\mathcal{O}$联系起来，为这些代数和群的一大类表示引入了张量积，证明其子范畴构成张量范畴并计算了融合规则。

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 in type $ε_0 +1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In fact we also obtain, for each $α< ε_0$, a finitely generated elementary amenable subgroup of $F$ whose EA-class is $α+ 2$. These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with $\mathbf{Z} + \mathbf{Z}$, $\mathbf{Z} \wr \mathbf{Z}$, a…

R. Thompson群与Cuntz代数的原子表示

Atomic representations of R. Thompson's groups and Cuntz's algebra

本文研究了Richard Thompson群的毕达哥拉斯酉表示，证明其可分解为扩散部分和原子部分，并完整描述了由抛物线子群族不可约单项表示构成的原子部分。

R. Thompson 群与 Cuntz 代数的不可约 Pythagorean 表示

Irreducible Pythagorean representations of R. Thompson's groups and of the Cuntz algebra

本文提出 Pythagorean 维数，用有限维线性代数函子式分类 Thompson 群与 Cuntz 代数的 Pythagorean 表示，并刻画其不可约模空间及与单项表示的关系。

Automorphisms of the generalised Thompson's group $T_{n,r}$

The recent paper "The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups $G_{n,r}$" of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterises the automorphisms of the Higman-Thompson groups $G_{n,r}$ as the specific subgroup of the rational group $\mathcal{R}_{n,r}$ of Grigorchuk, Nekrashevych and Suchanski{\u i}'s consisting of those elements which have the additional property of being bi-synchronizing. In this article, we extend the arguments of BCMNO and characterise the automorphism group of $T_{n,r}$ as a subgroup of …

$PL^+[0,1]$ 的有限生成子群的恰无限商

Just infinite quotients of finitely generated subgroups of $PL^+[0,1]$

本文证明 Richard Thompson 群 F 及两类分段同胚群的有限生成子群的恰无限商实际上交换，从而回答 Grigorchuk 的问题，并给出众多非顺从群例子。

森林skein群 III：单性

Forest-skein groups III: simplicity

本文研究森林skein群的导子群何时为单群，给出动力学与范畴论两种刻画，并构造关联于有限二叉树及n元Higman–Thompson群的多类单群例子。

Classification of Thompson related groups arising from Jones technology I

In the quest in constructing conformal field theories (CFT) Jones has discovered a beautiful and deep connection between CFT, Richard Thompson's groups and knot theory. This led to a powerful functorial framework for constructing actions of particular groups arising from categories such as Thompson's groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product where the largest Thompson's group $V$ is acting. These semidirect products have remarkable diagrammatic descriptions which were previously used to provide new examples of grou…

A finitely presented group of piecewise projective homeomorphisms

In this article we will describe a finitely presented subgroup of Monod's group of piecewise projective homeomorphisms of R. This in particular provides a new example of a finitely presented group which is nonamenable and yet does not contain a nonabelian free subgroup. It is in fact the first such example which is torsion free. We will also develop a means for representing the elements of the group by labeled tree diagrams in a manner which closely parallels Richard Thompson's group F.

分形的重排群

Rearrangement groups of fractals

本文为边替换系统构造了推广汤普森群的重排群，证明其适当地作用于局部有限CAT(0)立方复形，并据此证明某些重排群具有F无穷类型。

Classification of Thompson related groups arising from Jones technology II

In this second article, we continue to study classes of groups constructed from a functorial method due to Vaughan Jones. A key observation of the author shows that these groups have remarkable diagrammatic properties that can be used to deduce their properties. Given any group and two of its endomorphisms, we construct a semidirect product. In our first article dedicated to this construction, we classify up to isomorphism all these semidirect products when one of the endomorphisms is trivial and described their automorphism group. In this article we focus on the case where both endomorphisms …

On Groups of PL-homeomorphisms of the Real Line

Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a finite presentation with 2 generators and 2 relators, and a derived group that is simple; it admits a peculiar infinite presentation and has a local definition which implies that F is dense in the topological group of all orientation preserving homeomorphisms of the unit interval. In this monograph groups G are studied which depend on three parameters I, A, and P and which generalize the local definition of Thompson's group F thus: G consists of all orientation preserving PL-homeomorphisms of the rea…

有限呈现的简单群无分段投影作用

Finitely presented simple groups with no piecewise projective actions

本文构造了一类有限呈现的简单群，它们能忠实作用于圆，但无法通过分段仿射或分段投影方式作用于射影线。

Random sampling of trivials words in finitely presented groups

We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann distribution \begin{align*}π(w) &= (|w|+1)^α β^{|w|} \cdot Z^{-1} \end{align*} where $|w|$ is the length of a word $w$, $α$ and $β$ are parameters of the algorithm, and $Z$ is a normalising constant. It follows that words of the same length are sampled with the same probability. The distribution can be expressed in terms of the cogrowth series of the group, which…