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Starting from the work by Jones on representations of Thompson's group F , subgroups of F with interesting properties have been defined and studied. One of these subgroups is called the 3-colorable subgroup F , which consists of elements whose "regions" given by their tree diagrams are 3-colorable. On the other hand, in his work on representations, Jones also gave a method to construct knots and links from elements of F. Therefore it is a natural question to explore a relationship between elements in F and 3-colorable links in the sense of knot theory. In this paper, we show that all elements in F give 3-colorable links. [ABSTRACT FROM AUTHOR]

We will give a general criterion - the existence of an F-obstruction - for showing that a subgroup of PL+I does not embed into Thompson's group F. An immediate consequence is that Cleary's "golden ratio" group Ft does not embed into F. Our results also yield a new proof that Stein's groups Fp,q do not embed into F, a result first established by Lodha using his theory of coherent actions. We develop the basic theory of F-obstructions and show that they exhibit certain rigidity phenomena of independent interest. In the course of establishing the main result of the paper, we prove a dichotomy theorem for subgroups of PL+I. In addition to playing a central role in our proof, it is strong enough to imply both Rubin's Reconstruction Theorem restricted to the class of subgroups of PL+I and also Brin's Ubiquity Theorem. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F. Defining primes in a non-abelian monoid is highly non-trivial, which relies on a concept called "castling". Three types of castlings are essential to grasp the arithmetics. The divisor function τ on Thompson's monoid S satisfies τ(uv) ≤ τ(u)τ(v) for any u , v ∈ S . Then the limit τ 0 (u) = lim n → ∞ (τ (u n)) 1 / n exists. The quantity Ç (S) = sup 1 ≠ u ∈ S τ 0 (u) / τ (u) describes the complexity for castlings in S . We show that Ç (S) = 1 . Moreover, the Möbius function on S is calculated. And the Liouville function Ω on S is studied. [ABSTRACT FROM AUTHOR]

The set of finitely generated subgroups of the group PL_+(I) of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R. Thompson's group F. Here, we show that every finitely generated subgroup G

We prove that the group of almost-automorphisms of the infinite rooted regular d -ary tree 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 ≤ Aut (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 ∞ . Namely, we find some natural conditions on subgroups of G to ensure that V d (G) is of type F ∞ and, in particular, we prove this for all G in the infinite family of Šunić groups. We also prove that if G is itself of type F ∞ , then so is V d (G) , and that every finitely generated virtually free group is self-similar, so in particular every finitely generated virtually free group G yields a type F ∞ Röver–Nekrashevych group V d (G). [ABSTRACT FROM AUTHOR]

We address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if H and K are two non-solvable groups then a faithful Cl,τ action of H × K on a compact interval is not overlapping for all τ > 0, which by definition means that there must be non-trivial h ∈ H and k ∈ K with disjoint support. As a corollary we prove that the right-angled Artin group (F2 × F2) ∗ Z has critical regularity one, which is to say that it admits a faithful Cl action on I, but no faithful Cl,τ action. This is the first explicit example of a group of exponential growth which is without nonabelian subexponential growth subgroups, whose critical regularity is finite, achieved, and known exactly. Another corollary we get is that Thompson's group F does not admit a faithful Cl overlapping action on I, so that F ∗ Z is a new example of a locally indicable group admitting no faithful Cl action on I. [ABSTRACT FROM AUTHOR]

We study 2-generated subgroups $\langle f,g\rangle such that $\langle f^{2},g^{2}\rangle$ is isomorphic to Thompson's group $F$ , and such that the supports of $f$ and $g$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with non-abelian free subgroups, examples which do not admit faithful actions by $C^{2}$ diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $F$ as part of a general phenomenon among subgroups of $\operatorname{Homeo}^{+}(I)$. [ABSTRACT FROM AUTHOR]

The word problem for Thompson's group F has a solution, but it remains unknown whether F is automatic or has a finite or regular convergent (terminating and confluent) rewriting system. We show that the group F admits a natural extension of these two properties, namely autostackability, and we give an explicit bounded regular convergent prefix-rewriting system for F. [ABSTRACT FROM AUTHOR]

We introduce the Pythagorean C*-algebras and use the category/functor method to construct unitary representations of Thompson's groups from representations of them. We calculate several examples. [ABSTRACT FROM AUTHOR]

We present new results regarding automatic continuity, unifying some diagonalization concepts that have been developed over the years. For example, any homomorphism from a completely metrizable topological group to Thompson's group F has open kernel. A similar claim holds when F is replaced with a Baumslag-Solitar group or a torsion-free word hyperbolic group. [ABSTRACT FROM AUTHOR]

The scale of an endomorphism of a totally disconnected, locally compact group G is defined and an example is presented which shows that the scale function is not always continuous with respect to the Braconnier topology on the automorphism group of G. Methods for computing the scale, which is a positive integer, are surveyed and illustrated by applying them in diverse cases, including when G is compact; an automorphism group of a tree; Neretin's group of almost automorphisms of a tree; and a p-adic Lie group. The information required to compute the scale is reviewed from the perspective of the, as yet incomplete, general theory of totally disconnected, locally compact groups. [ABSTRACT FROM AUTHOR]

The space of configurations of n ordered points in the plane serves as a classifying space for the pure braid group PBn. Elements of Thompson's group F admit a model similar to braids, except instead of braiding the strands split and merge. In Belk's thesis, a space CF was considered of configurations of points on the real line allowing for splitting and merging, and a sketch of a proof was given that CF is a classifying space for F. The idea there was to build the universal cover and construct an explicit contraction to a point. However, this was never written up rigorously. Here we start with an established CAT(0) cube complex X on which F acts freely, and construct an explicit homotopy equivalence between X/F and CF, proving that CF is indeed a K(F, 1). [ABSTRACT FROM AUTHOR]

Let G be a locally compact group. Then for every G -space X the maximal G -proximity β G can be characterized by the maximal topological proximity β as follows: A β G ‾ B ⇔ ∃ V ∈ N e V A β ‾ V B. Here, β G : X → β G X is the maximal G -compactification of X (which is an embedding for locally compact G by a classical result of J. de Vries), V is a neighbourhood of e and A β G ‾ B means that the closures of A and B do not meet in β G X. Note that the local compactness of G is essential. This theorem comes as a corollary of a general result about maximal U -uniform G -compactifications for a useful wide class of uniform structures U on G -spaces for not necessarily locally compact groups G. It helps, in particular, to derive the following result. Let (U 1 , d) be the Urysohn sphere and G = Iso (U 1 , d) is its isometry group with the pointwise topology. Then for every pair of subsets A , B in U 1 , we have A β G ‾ B ⇔ ∃ V ∈ N e d (V A , V B) > 0. More generally, the same is true for any ℵ 0 -categorical metric G -structure (M , d) , where G : = A u t (M) is its automorphism group. [ABSTRACT FROM AUTHOR]

We determine exactly which graph products, also known as Right Angled Artin Groups, embed into Richard Thompson's group . It was shown by Bleak and Salazar-Díaz that 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. [ABSTRACT FROM AUTHOR]

We use a result of Kauffman to explore the connection between Richard Thompson's Group F and the four-color theorem. [ABSTRACT FROM AUTHOR]

Hughes has defined a class of groups that we call finite similarity structure (FSS) groups. Each FSS group acts on a compact ultrametric space by local similarities. The best-known example is Thompson’s group V. Guided by previous work on Thompson’s group, we show that many FSS groups are of type F∞. This generalizes work of Ken Brown from the 1980s. [ABSTRACT FROM PUBLISHER]

Guba [8] proved that the Dehn function of the Thompson group F is quadratic. In this paper, we apply his idea and extend his result to the generalized Thompson group F ( L ) for any integer L ≥ 3 . [ABSTRACT FROM AUTHOR]

In this note we show that the members of a certain class of local similarity groups are ${l}^{2}$-invisible, i.e. the (non-reduced) group homology of the regular unitary representation vanishes in all degrees. This class contains groups of type ${F}_{\infty }$, e.g. Thompson’s group $V$ and Nekrashevych–Röver groups. They yield counterexamples to a generalized zero-in-the-spectrum conjecture for groups of type ${F}_{\infty }$. [ABSTRACT FROM PUBLISHER]

Hassler Whitney's theorem of 1931 reduces the task of finding proper, vertex 4-colorings of triangulations of the 2-sphere to finding such colorings for the class ℌ of triangulations of the 2-sphere that have a Hamiltonian circuit. This has been used by Whitney and others from 1936 to the present to find equivalent reformulations of the 4 Color Theorem (4CT). Recently there has been activity to try to use some of these reformulations to find a shorter proof of the 4CT. Every triangulation in ℌ has a dual graph that is a union of two binary trees with the same number of leaves. Elements of a group known as Thompson's group F are equivalence classes of pairs of binary trees with the same number of leaves. This paper explores this resemblance and finds that some recent reformulations of the 4CT are essentially attempting to color elements of ℌ using expressions of elements of F as words in a certain generating set for F. From this, we derive information about not just the colorability of certain elements of ℌ, but also about all possible ways to color these elements. Because of this we raise (and answer some) questions about enumeration. We also bring in an extension E of the group F and ask whether certain elements "parametrize" the set of all colorings of the elements of ℌ that use all four colors. [ABSTRACT FROM AUTHOR]

We produce a sequence of markings S of Thompson's group F within the space $${{\mathcal G}_n}$$ of all marked n-generator groups so that the sequence ( F, S) converges to the free group on n generators, for n ≥ 3. In addition, we give presentations for the limits of some other natural (convergent) sequences of markings to consider on F within $${{\mathcal G}_3}$$, including ( F, { x, x, x}) and $${(F,\{x_0,x_1,x_0^n\})}$$. [ABSTRACT FROM AUTHOR]

We give criteria for determining the approximate length of elements in any given cyclic subgroup of the Thompson-Stein groups F(n1,...,nk) such that n1 - 1|ni - 1 ∀i ∈ {1,...,k} in terms of the number of leaves in the minimal tree-pair diagram representative. This leads directly to the result that cyclic subgroups are quasi-isometrically embedded in the Thompson-Stein groups. This result also leads to the corollaries that ℤn is also quasi-isometrically embedded in the Thompson-Stein groups for all n ∈ ℕ and that the Thompson-Stein groups have infinite dimensional asymptotic cone. [ABSTRACT FROM AUTHOR]

Abstract: We develop combinatorial methods for establishing lower bounds on the rotation distance between binary trees, i.e., equivalently, on the flip distance between triangulations of a polygon. These methods lead to sharp estimates for certain particular pairs of trees. As an application, we prove that, for each n, there exist size n trees at distance , i.e., the diameter of the nth associahedron has at least this value. [Copyright &y& Elsevier]

Thompson's group V can be thought of as the group of automorphisms of a certain algebra or as a subgroup of the group of self homeomorphisms of the Cantor set. Thus, the dynamics of an element of V can be studied. We analyze the dynamics in detail and use the analysis to give another solution of the conjugacy problem in V. [ABSTRACT FROM AUTHOR]

We introduce a new method for computing the word length of an element of Thompson's group F with respect to a "consecutive" generating set of the form Xn = {x0,x1, ...,xn}, which is a subset of the standard infinite generating set for F. We use this method to show that (F, Xn) is not almost convex, and has pockets of increasing, though bounded, depth dependent on n. [ABSTRACT FROM AUTHOR]

Amenability of some discrete subgroups of the group of diffeomorphisms ${\rm Diff}^3_+([0,1])$ of interval is proved. As a consequence, a solution of the problem of amenability of the Thompson group F is given. [ABSTRACT FROM AUTHOR]

Abstract: We produce two separate algebraic descriptions of the isomorphism classes of the solvable subgroups of the group of piecewise-linear orientation-preserving homeomorphisms of the unit interval under the operation of composition, and also of the generalized R. Thompson groups . The first description is as a set of isomorphism classes of groups which is closed under three algebraic operations, and the second is as the set of isomorphism classes of subgroups of a countable collection of easily described groups. We show the two descriptions are equivalent. [Copyright &y& Elsevier]

Rotation distance measures the difference in shape between binary trees of the same size by counting the minimum number of rotations needed to transform one tree to the other. We describe several types of rotation distance where restrictions are put on the locations where rotations are permitted, and provide upper bounds on distances between trees with a fixed number of nodes with respect to several families of these restrictions. These bounds are sharp in a certain asymptotic sense and are obtained by relating each restricted rotation distance to the word length of elements of Thompson's group F with respect to different generating sets, including both finite and infinite generating sets. [ABSTRACT FROM AUTHOR]

A subgroup of any group is called conjugately dense if it has nonempty intersection with each class of conjugate elements of the group. The aim of this paper is to prove the following. Let K be a locally finite field and H be an irreducible conjugately dense subgroup of the intermediate group SL3(K) ≤ G ≤ GL3(K); then H = G. This result confirms part of P. Neumann's conjecture from problem 6.38 in "Kourovka Notebook" for the group GL3(K) over locally finite field K. [ABSTRACT FROM AUTHOR]

We introduce forest diagrams to represent elements of Thompson's group F. These diagrams relate to a certain action of F on the real line in the same way that tree diagrams relate to the standard action of F on the unit interval. Using forest diagrams, we give a conceptually simple length formula for elements of F with respect to the {x0,x1} generating set, and we discuss the construction of minimum-length words for positive elements. Finally, we use forest diagrams and the length formula to examine the structure of the Cayley graph of F. [ABSTRACT FROM AUTHOR]

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. 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 moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of $(3+\sqrt5)/2$. [ABSTRACT FROM AUTHOR]

Although it is well known that the growth of Thompson's group F is exponential, the exact growth function is still unknown. Elements of its submonoid of positive words can be described using a binary rooted tree, whose norm can be computed assigning weights to each caret. Combining this fact with a combinatorial argument, the growth function of the submonoid is computed and thus providing a first step in the computation of the growth function of the group, as well as a lower bound for the growth rate for the group. [ABSTRACT FROM AUTHOR]

Any diagram group over a finite semigroup presentation acts properly, freely, and cellularly by isometrices on a proper CAT(0) cubical complex.The existence of a proper, cellular action by isometries on a CAT(0) cubical complex has powerful consequences for the acting group G. One gets, for example, a proof that G satisfies the Baum–Connes conjecture.Any diagram group over a finite presentation of a finite semigroup is of type F∞. [Copyright &y& Elsevier]