The distance from the origin in the word metric for generalizations F(p) of Thompson's group F is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of F(p). This interpretation of the metric is used to prove that every F(p) admits a quasi-isometric embedding into every F(q), and also to study the behavior of the shift maps under these embeddings.
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.
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.
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 $n \geq 3$.
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.
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 $H_{n,p}$.
Let $m\leq n\in \mathbb{N}$, and $G\leq S_m$ and $H\leq S_n$. 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 $V_n(H)$ in $V_m(G)$ via topological conjugation. With the same modular condition we also generalise a purely algebraic construction of Birget from 2019 to find a group $H\leq S_m$ and an embedding of $V_m(G)$ in $V_n(H)$.
We use geometric techniques to investigate several examples of quasi-isometrically embedded subgroups of Thompson's group F. Many of these are explored using the metric properties of the shift map phi in F. These subgroups have simple geometric but complicated algebraic descriptions. We present them to illustrate the intricate geometry of Thompson's group F as well as the interplay between its standard finite and infinite presentations. These subgroups include those of the form F^m cross Z^n, for integral non-negative m and n, which were shown to occur as quasi-isometrically embedded subgroups…
We obtain a characterisation of confined subgroups of Thompson's group $F$. As a result, we deduce that orbital graph of a point under 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.
Pursueing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group $T^*$ (and its further generalizations) which is an extension of the Ptolemy-Thompson group $T$ by means of the full braid group $B_{\infty}$ on infinitely many strands. We prove that it is a finitely presented group with solvable word problem, and give an explicit presentation of it.
Lehnert and Schweitzer show in [20] that R. Thompson's group $V$ is a co-context-free ($co\mathcal{CF}$) group, thus implying that all of its finitely generated subgroups are also $co\mathcal{CF}$ groups. Also, Lehnert shows in his thesis that $V$ embeds inside the $co\mathcal{CF}$ group $\mathrm{QAut}(\mathcal{T}_{2,c})$, which is a group of particular bijections on the vertices of an infinite binary $2$-edge-colored tree, and he conjectures that $\mathrm{QAut}(\mathcal{T}_{2,c})$ is a universal $co\mathcal{CF}$ group. We show that $\mathrm{QAut}(\mathcal{T}_{2,c})$ embeds into $V$, and thus …
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.
We present a generalization of the Dehornoy-Brin braided Thompson group $BV_2$ that uses recursive braids. Our new groups are denoted by $BV_{n,r}(H)$, for all $n\geq 2,r\geq 1$ and $H \leq \mathcal{B}_n$, where $\mathcal{B}_n$ is the braid group on $n$ strands. We give a new approach to deal with braided Thompson groups by using strand diagrams. We show that $BV_{n,r}(H)$ is finitely generated if $H$ is finitely generated.
In a recent paper, we showed that groups admitting "veelike actions" on a finite language embed in mapping class groups of certain two-sided subshifts. In this note, we illustrate this theorem for the embedding of Thompson's $V$ by exhibiting the piecewise linear local rules for the embedding. These turn out to split the embedding into the mapping class group, showing that $V$ even embeds in the mapping class group by homeomorphisms.
Let $p$ be a prime and $\mathcal{F}$ be a saturated fusion system over a finite $p$-group $P$. The fusion system $\mathcal{F}$ is said to be nilpotent if $\mathcal{F}=\mathcal{F}_{P}(P)$. We provide new criteria for a saturated fusion system $\mathcal{F}$ to be nilpotent, which may be viewed as extending the Glauberman-Thompson $p$-nilpotency criterion to fusion systems.
We show how to construct a family of groups with simple commutator subgroups from aperiodic 1-vertex, finitely aligned higher rank graphs (which are, in fact, a class of cancellative monoids). Inverse semigroups form the intermediary between these cancellative monoids and the family of groups we are interested in. These groups can naturally be viewed as higher-dimensional generalizations of the classical Thompson groups since the finite direct products of free monoids are examples of the appropriate 1-vertex higher rank graphs.
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 construct a family of groups from suitable higher rank graphs which are analogues of the finite symmetric groups. We introduce homological invariants showing that many of our groups are, for example, not isomorphic to $nV$, when $n \geq 2$.
We prove that every countable left-ordered group embeds into a finitely generated left-ordered simple group. Moreover, if the first group has a computable left-order, then the simple group also has a computable left-order. We also obtain a Boone-Higman-Thompson type theorem for left-orderable groups with recursively enumerable positive cones. These embeddings are Frattini embeddings, and isometric whenever the initial group is finitely generated. Finally, we reprove Thompson's theorem on word problem preserving embeddings into finitely generated simple groups and observe that the embedding is …
The group of $\mathcal C^1$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher dimensional generalizations $nV$ of Thompson's group $V$ arise when we consider products of central ternary Cantor sets. We derive that the $\mathcal C^2$-smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.
We prove that seesaw words exist in Thompson's Group F(N) for N=2,3,4,... with respect to the standard finite generating set X. A seesaw word w with swing k has only geodesic representatives ending in g^k or g^{-k} (for given g\in X) and at least one geodesic representative of each type. The existence of seesaw words with arbitrarily large swing guarantees that F(N) is neither synchronously combable nor has a regular language of geodesics. Additionally, we prove that dead ends (or k--pockets) exist in F(N) with respect to X and all have depth 2. A dead end w is a word for which no geodesic pat…
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]).
Let $G$ be a finitely generated group, and let $Σ$ be a finite subset that generates $G$ as a monoid. The \emph{word problem of $G$ with respect to $Σ$} consists of all words in the free monoid $Σ^{\ast}$ that are equal to the identity in $G$. The \emph{co-word problem of $G$ with respect to $Σ$} is the complement in $Σ^{\ast}$ of the word problem. We say that a group $G$ is \emph{co$\mathcal{CF}$} if its co-word problem with respect to some (equivalently, any) finite generating set $Σ$ is a context-free language. We describe a generalized Thompson group $V_{(G, θ)}$ for each finite group G an…
The purpose of this paper is to study the properties of the irrational-slope Thompson's group $F_τ$ introduced by Cleary in 1995. We construct presentations, both finite and infinite and we describe its combinatorial structure using binary trees. We show that its commutator group is simple. Finally, inspired by the case of Thompson's group F, we define a unique normal form for the elements of the group and study the metric properties for the elements based on this normal form. As a corollary, we see that several embeddings of $F$ in $F_τ$ are undistorted.
In this note, a generalization of the Thompson transfer lemma and its various extensions, most recently due to Lyons, is proven in the context of saturated fusion systems. A strengthening of Alperin's fusion theorem is also given in this setting, following Alperin's own "up and down" fusion.
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…
Orderable Thompson-like groups arising from Ore categories
Université de Genève（日内瓦大学）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Laboratoire de mathématiques d’Orsay（奥赛数学实验室）；Université de Rennes（雷恩大学）；IRMAR（雷恩数学与应用研究所）
The purpose of this article is prove that Thompson's group F is amenable. The methods developed will then be used to prove a generalization of Hindman's theorem for the free nonassociative binary system on one generator.
In 1984 Brown and Geoghegan proved that Thompson's group $F$ is of type $\textrm{F}_\infty$, making it the first example of an infinite dimensional torsion-free group of type $\textrm{F}_\infty$. Over the decades a different, shorter proof has emerged, which is more streamlined and generalizable to other groups. It is difficult, however, to isolate this proof in the literature just for $F$ itself, with no complicated generalizations considered and no additional properties proved. The goal of this expository note then is to present the "modern" proof that $F$ is of type $\textrm{F}_\infty$, and…
The purpose of this article is to formulate conjectural generalizations of Hindman's Theorem and Ellis's Lemma for nonassociative binary systems and relate them to the amenability problem for Thompson's group $F$. Partial results are obtained for both conjectures. The paper will also contain some general analysis of the conjectures.
In this paper, we will study presentations of the continuous full group $Γ_A$ of a one-sided topological Markov shift $(X_A,σ_A)$ for an irreducible matrix $A$ with entries in $\{0,1\}$ as a generalization of Higman-Thompson groups $V_N, 1<N \in {\mathbb{N}}$. We will show that the group $Γ_A$ can be represented as a group $Γ_A^{\operatorname{tab}}$ of matrices, called $A$-adic tables, with entries in admissible words of the shift space $X_A$, and a group $Γ_A^{\operatorname{PL}}$ of right continuous piecewise linear functions, called $A$-adic PL functions, on $[0,1]$ with finite singularities…
We give first examples of finitely generated groups having an intermediate, with values in (0,1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert space compression of Richard Thompson's group $F$ is equal to 1/2, the Hilbert space compression of the restricted wreath product $Z\wr Z$ is between 1/2 and 3/4, and the Hilbert space compression of $Z\wr (Z\wr Z)$ is between 0 and 1/2. In general, we find a relationship between…
We show that locally solvable subgroups of PLo(I) are countable. Then for each countable ordered set, we construct a locally solvable subgroup of Thompson's Group F. We develop machinery for understanding embeddings from solvable subgroups into solvable subgroups. Finally, we apply this machinery to show the ordered sets used in our construction are invariant under isomorphisms between the groups constructed. Therefore, we effectively distinguish the groups and provide uncountably many non-isomorphic locally solvable, hence elementary amenable, subgroups of Thompson's Group F.
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 …
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 answer the question by Matt Brin on the structure of diagram groups over semigroup presentation ${\mathcal P}=\langle a,b,c\mid a=bc,b=ca,c=ab\rangle$. In the talk on Oberwolfach workshop, Brin conjectured that the diagram group over $\mathcal P$ with base $a$ is isomorphic to the generalized Thompson's group $F_9$. We confirm this conjecture and consider some generalizations of this fact.
We prove a variety of results about the group von Neumann algebras associated to Thompson-like groups arising from so called $d$-ary cloning systems. Cloning systems are a framework developed by Witzel and the second author, with a $d$-ary version subsequently developed by Skipper and the second author, which can be used to construct generalizations of the classical Thompson's groups $F$, $T$, and $V$. Given a family of groups $(G_n)_{n\in\mathbb{N}}$ with a $d$-ary cloning system, we get a Thompson-like group $\mathscr{T}_d(G_*)$, and in this paper we find some mild, natural conditions under …
We observe that the group of all lifts of elements of Thompson's group $T$ to the real line is finitely presented and contains the additive group $\mathbb{Q}$ of the rational numbers. This gives an explicit realization of the Higman embedding theorem for $\mathbb{Q}$, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.
We prove that random walks on Thompson's group $F$ driven by strictly non-degenerate finitely supported probability measures $μ$ have a non-trivial Poisson boundary. The proof consists in an explicit construction of two different non-trivial $μ$-boundaries. Both of them are defined in terms of the Schreier graph $Γ$ on the dyadic-rational orbit of the canonical action of $F$ on the unit interval (actually, we consider a natural embedding of $F$ into the group $PLF({\mathbb R})$ of piecewise linear homeomorphisms of the real line, and realize $Γ$ on the dyadic-rational orbit in ${\mathbb R}$). …
We set up a Grothendieck spectral sequence which generalizes the Lyndon--Hochschild--Serre spectral sequence for a group extension $K\mono G\epi Q$ by allowing the normal subgroup $K$ to be replaced by a subgroup, or family of subgroups which satisfy a weaker condition than normality. This is applied to establish a decomposition theorem for certain groups as fundamental groups of graphs of Poincaré duality groups. We further illustrate the method by proving a cohomological vanishing theorem which applies for example to Thompson's group $F$.
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 …
This manuscript represents the author's PhD dissertation thesis.The first part studies decision problems in Thompson's groups F,T,V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson's group F and suitable larger groups of piecewise-linear homeomorphisms of the unit interval. We determine algorithms to compute roots and centralizers in these groups and to detect periodic points and their behavior by looking at a particular diagram associated to an element. In the second part, we describe the structure of subgroups of the group of all homeomor…
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.
We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov's notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum-Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.
