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 $\mathrm{Aut}{G_{n,r}}$. We then show that the groups $\mathrm{Out}{T_{n,r}}$ can be identified with subgroups of the group $\mathrm{Out}{T_{n,n-1}}$. Extending results of Brin and Guzman, we show that the groups $\mathrm{Out}{T_{n,r}}$, for $n>2$, are all infinite and contain an isomorphic copy of Thompson's group $F$. For $X \in \{T,G\}$, we study the groups $\mathrm{Out}{X_{n,r}}$ and show that these fit in a lattice structure where $\mathrm{Out}{X_{n,1}} \unlhd \mathrm{Out}{X_{n,r}}$ for all $1 \le r \le n-1$ and $\mathrm{Out}{X_{n,r}} \unlhd \mathrm{Out}{X_{n,n-1}}$. This gives a partial answer to a question in BCMNO concerning the normal subgroup structure of $\mathrm{Out}{G_{n,n-1}}$. Furthermore, we deduce that for $1\le j,d \le n-1$ such that $d = \gcd(j, n-1)$, $\mathrm{Out}{X_{n,j}} = \mathrm{Out}{X_{n,d}}$ extending a result of BCMNO for the groups $G_{n,r}$ to the groups $T_{n,r}$. We give a negative answer to the question in BCMNO which asks whether or not $\mathrm{Out}{G_{n,r}} \cong \mathrm{Out}{G_{n,s}}$ if and only if $\gcd(n-1,r) = \gcd(n-1,s)$. We conclude by showing that the groups $T_{n,r}$ have the $R_{\infty}$ property extending the result of Burillo, Matucci and Ventura and, independently, Gon{\c c}alves and Sankaran, for Thompson's group $T$.
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.
We look at the automorphisms of Thompson type groups of piecewise linear homeomorphisms of the real line or circle that use slopes that are integral powers of a fixed integer n with n>2. We show that large numbers of "exotic" automorphisms appear---automorphisms that are represented as conjugation by non-PL homeomorphisms of the real line or circle. This is in contrast to the n=2 case where no such automorphisms appear.
Golan and Sapir \cite{MR3978542} proved that the Thompson's groups $F$, $T$ and $V$ have linear divergence. In the current paper, we focus on the divergence properties of several generalisation of the Thompson's groups, we first consider the Brown-Thompson's groups $F_n$, $T_n$ and $V_n$ and found out that these groups also have linear divergence function. We then consider the braided Thompson's groups $BF$ and $\widehat{BF}$ and $\widehat{BV}$ together with the result in \cite{Kodama:2020to} we conclude that theses groups have linear divergence.
For an arbitrary sequence $(G_s)$ of subgroups of finite index in the generalised Thompson group $$F_{n, \infty} = \langle x_0, x_1, \ldots, x_m, \ldots \mid x_i^{x_j} = x_{i+ n-1} \hbox{ for } i > j \geq 0 \rangle$$ it is shown that $\sup_{s \geq 1} d(G_s) < \infty$ and that the deficiency gradient of $F_{n, \infty}$ with respect to $(G_s)$ is 0 provided $[G : G_s]$ tends to infinity. A higher dimensional analogue is considered for $n = 2$.
Brown has defined the generalised Thompson's group $F_n$, $T_n$, where $n$ is an integer at least $2$ and Thompson's groups $F= F_2$ and $T =T_2$ in the 80's. Burillo, Cleary and Stein have found that there is a quasi-isometric embedding from $F_n$ to $F_m$ where $n$ and $m$ are positive integers at least 2. We show that there is a quasi-isometric embedding from $T_n$ to $T_2$ for any $n \geq 2$ and no embeddings from $T_2$ to $T_n$ for $n \geq 3$.
Let $V_r(Σ)$ be the generalised Thompson group defined as the automorphism group of a valid, bounded, and complete Cantor algebra. We show that that for every $n>0$ there is a $k>n,$ such that there exists a $k$-dimensional $n$-connected simplicial complex $K$ such that $V_r(Σ)$ acts on $K$ with finite stabilisers. We also determine the number of conjugacy classes of finite cyclic subgroups of a given order $m$ in Brin-Thompson groups. We apply our computations to the rationalised Farrell-Jones assembly map in algebraic $K$-theory.
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 describe some of the geometric properties of the automorphism group Aut(F) of Thompson's group F. We give realizations of Aut(F) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F) and their metric properties. In particular, we show that the subgroup of inner automorphisms of F is at least quadratically distorted in Aut(F), whereas other subgroups of Aut(F) isomorphic to F are undistorted.
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…
We explain how to construct a morphism from the group of birational automorphisms of CP^2 preserving the logarithmic Poisson bracket to the Thompson group T. Than we give a linear representation of the former group, provide some information about Thompson group T and give it's new presentation in terms of generators and relations, and speculate about the presentation of the group of birational symplectomorphisms.
Let $1 \le r < n$ be integers. We give a proof that the group $\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, σ_{n}})$ of automorphisms of the one-sided shift on $n$ letters embeds naturally as a subgroup $\mathcal{H}_{n}$ of the outer automorphism group $\mathop{\mathrm{Out}}({G_{n,r}})$ of the Higman-Thompson group $G_{n,r}$. From this, we can represent the elements of $\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, σ_{n}})$ by finite state non-initial transducers admitting a very strong synchronizing condition. Let $H \in \mathcal{H}_{n}$ and write $|H|$ for the number of states of the minimal tran…
We show that for any full and sufficiently transitive (i.e. \textit{flexible}) group $G$ of homeomorphisms of Cantor space, $\mathrm{Aut}(\mathrm{Aut}(G)) = \mathrm{Aut}(G)$. This class contains many generalisations of the Higman-Thompson groups $G_{n,r}$, and the Rational group $\mathcal{R}_{2}$ of Grigorchuk, Nekrashevych, and Suchanski{\u ı}. We also demonstrate that for generalisations $T_{n,r}$ of R. Thompson's group $T$, $\mathrm{Aut}(\mathrm{Aut}(T_{n,r}))= \mathrm{Aut}(T_{n,r})$. In the case of the groups $G_{n,r}$ and $T_{n,r}$ our results extend results of Brin and Guzm{\' a}n for Th…
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.
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 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 $\mathbb{Z}/2\mathbb{Z}$.
We prove that the tree almost automorphism groups admit exactly three commensurability classes of closed commensurated subgroups. Our proof utilizes an independently interesting characterization of subgroups of the tree almost automorphism groups which contain only periodic elements in terms of the dynamics of the action on the boundary of the tree. Our results further cover several interesting finitely generated subgroups of the tree almost automorphism groups, including the Thompson groups $F$, $T$, and $V$. We show in particular that Thompson's group $T$ has no commensurated subgroups other…
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 $V$.
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 $H_{n,p}$.
We determine when two almost automorphisms of a regular tree are conjugate. This is done by combining the classification of conjugacy classes in the automorphism group of a level-homogeneous tree by Gawron, Nekrashevych and Sushchansky and the solution of the conjugacy problem in Thompson's $V$ by Belk and Matucci. We also analyze dynamics of tree almost automorphisms.
We classify a large class of small groups of finite Morley rank: $N_\circ^\circ$-groups which are the infinite analogues of Thompson's $N$-groups. More precisely, we constrain the $2$-structure of groups of finite Morley rank containing a definable, normal, non-soluble, $N_\circ^\circ$-subgroup.
A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs : abelian groups and generalised dicyclic groups. Indeed, any Cayley graph on such a group admits specific additional graph automorphisms that depend only on the group. Recently, Dobson and the last two authors showed that almost all Cayley graphs on abelian groups admit no automorphisms other than these obvious necessary ones. In this paper, we prove the analogous resul…
Let $p\ge 3$ be a prime. A generalised multi-edge spinal group is a subgroup of the automorphism group of a regular $p$-adic rooted tree T that is generated by one rooted automorphism and $p$ families of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalises the concepts of multi-edge spinal groups, including the widely studied GGS-groups, and the extended Gupta-Sidki groups that were introduced by Pervova. Extending techniques that were developed in these more special cases, we prove: generalised multi-edge spi…
In this short article, we prove that any automorphism of the R. Thompson's group $F$ has infinitely many twisted conjugacy classes. The result follows from the work of Matthew Brin, together with a standard facts on R. Thompson's group $F$, and elementary properties of the Reidemeister numbers.
A celebrated result of J. Thompson says that if a finite group $G$ has a fixed-point-free automorphism of prime order, then $G$ is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.
We study the properties of rotation numbers for some groups of piecewise linear homeomorphisms of the circle. We use these properties to obtain results on PL rigidity, non isomorphicity, non exoticity of automorphisms, non smoothability for Thompson-Stein groups.
In this article, we further explore the nature of a connection between the groups of automorphisms of full shift spaces and the groups of outer automorphisms of the Higman--Thompson groups $\{G_{n,r}\}$. We show that the quotient of the group of automorphisms of the (two-sided) shift dynamical system $\mathrm{Aut}(X_n^{\mathbb{N}}, σ_{n})$ by its centre embeds as a particular subgroup $\mathcal{L}_{n}$ of the outer automorphism group $\mathop{\mathrm{Out}}(G_{n,n-1})$ of $G_{n,n-1}$. It follows by a result of Ryan that we have the following central extension: $$\langle σ_{n}\rangle \hookrighta…
We show that Thompson's group F is the symmetry group of the "generic idempotent". That is, take the monoidal category freely generated by an object A and an isomorphism A \otimes A --> A; then F is the group of automorphisms of A.
We revisit our construction of the Thompson groups from the polycyclic inverse monoids in the light of new research. Specifically, we prove that the Thompson group $G_{n,1}$ is the group of units of a Boolean inverse monoid $C_{n}$ called the Cuntz inverse monoid. This inverse monoid is proved to be the tight completion of the polycyclic inverse monoid $P_{n}$. The étale topological groupoid associated with $C_{n}$ under non-commutative Stone duality is the usual groupoid associated with the corresponding Cuntz $C^{\ast}$-algebra. We then show that the group $G_{n,1}$ is also the group of auto…
The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their automorphism groups may be much larger than the groups of the corresponding schemes. We determine the parameters for which the graphs are connected, or equivalently, the schemes are primitive. Also we prove that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and we determine precisely the…
For a finite digraph $D$, we define the corresponding subshift of finite type $(X_D, σ_D)$ to be the dynamical system where $X_D$ is the set of all bi-infinite walks through $D$ and $σ_D$ is the shift operator. Two digraphs $D_1$ and $D_2$ are called shift equivalent if there is $m\geq 1$ such that $(X_{D_1}, σ_{D_1}^n)$ and $(X_{D_2}, σ_{D_2}^n)$ are topologically conjugate for all $n\geq m$. They are called strong shift equivalent if this holds for $m=1$. In this paper we introduce a new category which generalises the category of subshifts of finite type and topological conjugacy. Our catego…
We consider generalisations of Thompson's group $V$, denoted $V_r(Σ)$, which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, $V_r(Σ)$ is the full automorphism group of a Cantor-algebra. Under some further minor restrictions, we prove that these groups are of type $\mathrm{F}_\infty$ and that this implies that also centralisers of finite subgroups are of type $\mathrm{F}_\infty$.
Tent and Ziegler proved that the automorphism group of the Urysohn sphere is simple and that the automorphism group of the Urysohn space is simple modulo bounded automorphisms. A key component of their proof is the definition of a stationary independence relation (SIR). In this paper we prove that the existence of a SIR satisfying some extra axioms is enough to prove simplicity of the automorphism group of a countable structure. The extra axioms are chosen with applications in mind, namely homogeneous structures which admit a "metric-like amalgamation", for example all primitive 3-constrained …
The groups QF, QT, and QV are groups of quasi-automorphisms of the infinite binary tree. Their names indicate a similarity with Thompson's well-known groups F, T, and V. We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type F-infinity. Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on CAT(0) cubical complexes.
In a seminal paper, Brin demonstrates that the outerautomorphism group of Thompson group $T$ is isomorphic to the cyclic group of order two. In this article, building on characterisation of automorphisms of the Higman-Thompson groups $G_{n,r}$ and $T_{n,r}$ as groups of transducers, we give a new proof, automata theoretic in nature, of Brin's result. We also demonstrate that the group of outerautomorphisms of Thompson's group $V = G_{2,1}$ contains an isomorphic copy of Thompson's group $F$. This extends a result of the author demonstrating that whenever $n \ge 3$ and $1 \le r < n$ the outerau…
We study the group $QV$, the self-maps of the infinite $2$-edge coloured binary tree which preserve the edge and colour relations at cofinitely many locations. We introduce related groups $QF$, $QT$, $\tilde{Q}T$, and $\tilde{Q}V$, prove that $QF$, $\tilde{Q}T$, and $\tilde{Q}V$ are of type $\mathrm{F}_\infty$, and calculate finite presentations for them. We calculate the normal subgroup structure and rational homology of all $5$ groups, the Bieri--Neumann--Strebel--Renz invariants of $QF$, and discuss the relationship of all $5$ groups with other generalisations of Thompson's groups.
The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of profinite groups which generalises this phenomenon, and explore some consequences for the structure of profinite groups.
Motivated by classic theorems of Thompson and Berger on the Fitting height of finite groups with a fixed-point-free automorphism of coprime order, we conjecture that, for every non-zero polynomial $f(x) = a_0 + a_1 x + \cdots + a_d x^d \in \mathbb{Z}[x] $, there is an integer $k > 0$ with the following property. Let $G$ be a finite (solvable) group with a fixed-point-free automorphism $α$ satisfying $\gcd(|G|,k)= 1$ and $$\{ g^{a_0} \cdot α(g)^{a_1} \cdot α^2(g)^{a_2} \cdots α^d(g)^{a_d} | g \in G \} = \{1\}.$$ Then the Fitting height of $G$ is at most the number of irreducible factors of $f(x…
We show that representations of the Thompson group $F$ in the automorphisms of a noncommutative probability space yield a large class of bilateral stationary noncommutative Markov processes. As a partial converse, bilateral stationary Markov processes in tensor dilation form yield representations of $F$. As an application, and building on a result of Kümmerer, we canonically associate a representation of $F$ to a bilateral stationary Markov process in classical probability.
We establish compact presentability, i.e. the locally compact version of finite presentability, for an infinite family of tree almost automorphism groups. Examples covered by our results include Neretin's group of spheromorphisms, as well as the topologically simple group containing the profinite completion of the Grigorchuk group constructed by Barnea, Ershov and Weigel. We additionally obtain an upper bound on the Dehn function of these groups in terms of the Dehn function of an embedded Higman-Thompson group. This, combined with a result of Guba, implies that the Dehn function of the Nereti…
A generalised quadrangle is a point-line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given a line L and a point p not incident with L, there is a unique point on L collinear with p. They are a specific case of the generalised polygons introduced by Tits, and these structures and their automorphism groups are of some importance in finite geometry. An integral part of understanding the automorphism groups of finite generalised quadrangles is knowing which groups can act primitively on their points, and in particular, which almost simple groups arise as au…
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 investigate the relationship between endomorphisms of the Cuntz algebra ${\mathcal O}_2$ and endomorphisms of the Thompson groups $F$, $T$ and $V$ represented inside the unitary group of ${\mathcal O}_2$. For an endomorphism $λ_u$ of ${\mathcal O}_2$, we show that $λ_u(V)\subseteq V$ if and only if $u\in V$. If $λ_u$ is an automorphism of ${\mathcal O}_2$ then $u\in V$ is equivalent to $λ_u(F)\subseteq V$. Our investigations are facilitated by introduction of the concept of modestly scaling endomorphism of ${\mathcal O}_n$, whose properties and examples are investigated.
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 $T$…
We show that several classes of groups G of PL-homeomorphisms of the real line admit non-trivial homomorphisms from G to the additive group of reals that are fixed by every automorphism of G. The classes of groups enjoying the stated property include the generalisations of Thompson's group F studied by K. S. Brown, M. Stein, S. Cleary, and Bieri-Strebel but also the class of groups investigated by Bieri-Neumann-Strebel in [BNS87, Theorem 8.1]. It follows that every automorphism of a group in one of these classes has infinitely many associated twisted conjugacy classes.
We prove that the group of almost-automorphisms of the infinite rooted regular $d$-ary tree $\mathcal{T}_d$ arises naturally as the Thompson-like group of a so called $d$-ary cloning system. A similar phenomenon occurs for any Röver-Nekrashevych group $V_d(G)$, for $G\le Aut(\mathcal{T}_d)$ a self-similar group. We use this framework to expand on work of Belk and Matucci, who proved that the Röver group, using the Grigorchuk group for $G$, is of type $F_\infty$. Namely, we find some natural conditions on subgroups of $G$ to ensure that $V_d(G)$ is of type $F_\infty$, and in particular we prove…
Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes that can be constructed from centrally symmetric triangulations of even-sided polygons. In this article we introduce an infinite-dimensional analogue and prove that the group of symmetries of our construction is a semidirect product of a degree 2 central extension of Thompson's infinite finitely presented simple group T with the cyclic group of order 2. These results are inspired by a similar recent analysis by the first author of the automorphism group of an infinite-dimensional associahedron.
We prove that a permutation group in which different finite sets have different stabilizers cannot satisfy any group law. For locally compact topological groups with this property we show that almost all finite subsets of the group generate free subgroups. We derive consequences of these theorems on Thompson's group $F$, weakly branch groups, automorphism groups of regular trees and profinite groups with alternating composition factors of unbounded degree.
