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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 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]

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 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]

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]

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]