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 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 $H_{n,p}$.
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…
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.
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 …
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 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.
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 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 …
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.
We investigate the connection between Tamari lattices and the Thompson group F, summarized in the fact that F is a group of fractions for a certain monoid F+sym whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice operations are the counterparts of the least common multiple and greatest common divisor operations in F+sym. As an application, we show that, for every n, there exists a length l chain in the nth Tamari lattice whose endpoints are at distance at most 12l/n.
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 $H,G$.
By the density of a finite graph we mean its average vertex degree. For an $m$-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with $m$ generators is amenable iff the density of the corresponding Cayley graph equals $2m$. A famous problem on the amenability of R.\,Thompson's group $F$ is still open. What is known due to the result by Belk and Brown, is that the density of its Cayley graph in the standard set of group generators $\{x_0,x_1\}$, is at least $3.5$. This estimat…
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.
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.
The higher-dimensional Thompson groups $nV$, for $n \geq 2$, were introduced by Brin in 2005. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in $nV$ and reflecting the self-similar structure of $n$-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.
We show a homological result for the class of planar or symmetric operad groups: We show that under certain conditions, group (co)homology of such groups with certain coefficients vanishes in all dimensions, provided it vanishes in dimension $0$. This can be applied for example to $l^2$-homology or cohomology with coefficients in the group ring. As a corollary, we obtain explicit vanishing results for Thompson-like groups such as the Brin-Thompson groups $nV$.
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 …
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 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…
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.
We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pure braid groups. We give new examples of families of groups that admit a cloning system and study how the finiteness properties of the resulting generalized Thompson group depend on those of the original groups. The main new examples here include upper triangular matrix groups, mo…
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.
The "relating" entwines three problems: 1. Davenport's Problem, describing pairs of polynomials over Q whose ranges on Z/p are the same for almost all p. 2. Showing that the monodromy groups of rational function maps over the complexes are limited to a finite set of groups, outside of groups close to alternating groups (example, symmetric groups) with special representations, and dihedral and cyclic groups. 3. Relating the genus 0 modular curves to the character group of the Monster simple group, so-called Monstrous Moonshine. http://www.math.uci.edu/~mfried/pathlist-cov/thomp-genus0.html has …
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 propose a new unifying framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application, we proof a theorem which implies that planar or symmetric or braided operads with transformations satisfying some finiteness conditions yield operad groups of type $F_\infty$. This unifies and extends existing proofs that certain Thompson-like groups are of type $F_\infty$.
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 $T$…
In joint work of the author with Stefan Witzel, a procedure was developed for building new examples of groups in the extended family of R. Thompson's groups, using what we termed \emph{cloning systems}. These new Thompson-like groups can be thought of as limits of families of groups, though unlike other limiting processes, e.g., direct limits, these tend to be well behaved with respect to finiteness properties. In this expository note, we distill the crucial parts of that 50-page paper into a more digestible form, for those curious to understand the construction but less curious about the grit…
We consider a class of groups $V_n(G)$ which are supergroups of the Higman-Thompson groups $V_n$. These groups fit in a framework of Elizabeth Scott for generating infinite virtually simple groups, and the groups we study in particular are initially introduced by Farley and Hughes. The group $V_n(G)$ is the result one obtains by taking the $V_n$ generators and adding a tree automorphism for each generator of a subgroup $G$ of the symmetric group on $n$ letters, where the new generators each permute the child leaves of a specific vertex $α$ of the infinite rooted $n$-ary tree according to the p…
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 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 study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thus extend earlier rigidity results about quasi-isometric embeddings to the setting of semisimple Lie groups. We also present some examples when the rigidity does not hold, including first examples in which every flat is mapped into multiple flats.
We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric embeddings of higher rank symmetric spaces. In particular, we produce embeddings of $SL(n,\mathbb R)$ into $Sp(2(n-1),\mathbb R)$ when no isometric embeddings exist. A key ingredient in our proofs of rigidity results is a direct generalization of the Mostow-Morse Lemma in higher rank. Typically this lemma is replaced by the quasi-flat theorem which says that ma…
Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasi-isometric embeddings when the control functions are not necessarily affine. In this paper, we will be particularly interested in coarse embeddings between symmetric spaces and Euclidean buildings. The quasi-isometric case is very well understood thanks to the rigidity results for symmetric spaces and buildings of higher rank by Anderson-Schroeder, Kleiner, Kleiner-Leeb, Eskin-Farb and Fisher-Whyte. In particular, it is well known that the rank of these spaces is monotonous under quasi-isometric embeddings. This i…
