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.
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}.
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.
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…
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 $Φ_T(n)\preceq n^5$.
Richard Thompson's group F is the group of piecewise linear homeomorphisms of the unit interval with a finite number of break points, all at dyadic rational numbers (their denominators are powers of 2) and with slopes which are powers of 2. A discrete group G is amenable if there exists a finitely-additive probability measure on G which is invariant under left translations and is defined on all subsets of G. The amenability question for F is a well known open problem. In this paper we prove that group F is not amenable.
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.
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.
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…
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…
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…
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…
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.
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.
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 $p$.
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…
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.
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.
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.
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…
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 …
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.
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.
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…
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…
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 …
We say that $x,y\in Γ$ are in the same $ϕ$-twisted conjugacy class and write $x\sim_ϕy$ if there exists an element $γ\in Γ$ such that $y=γxϕ(γ^{-1})$. This is an equivalence relation on $Γ$ called the $ϕ$-twisted conjugacy. Let $R(ϕ)$ denote the number of $ϕ$-twisted conjugacy classes in $Γ$. If $R(ϕ)$ is infinite for all $ϕ\in Aut(Γ)$, we say that $Γ$ has the $R_\infty$-property. The purpose of this note is to show that the symmetric group $S_\infty$, the Houghton groups and the pure symmetric automorphism groups have the $R_\infty$-property. We show, also, that the Richard Thompson group $T$…
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…
