Narrow Results

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.

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.

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 X_n = {x₀,x₁, …,x_n}, which is a subset of the standard infinite generating set for F. We use this method to show that (F, X_n) is not almost convex, and has pockets of increasing, though bounded, depth dependent on n.

We give criteria for determining the approximate length of elements in any given cyclic subgroup of the Thompson–Stein groups F(n₁,…,n_k) such that n₁ - 1|n_i - 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.

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.

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 $G\le \mathrm{\text{Aut}}({{𝒯}}_d)$ 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 ${\mathrm{\text{F}}}_{\infty }$. Namely, we find some natural conditions on subgroups of $G$ to ensure that $V_d(G)$ is of type ${\mathrm{\text{F}}}_{\infty }$ 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 ${\mathrm{\text{F}}}_{\infty }$, 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 ${\mathrm{\text{F}}}_{\infty }$ Röver–Nekrashevych group $V_d(G)$.