Thompson's Group v From a Dynamical Viewpoint

Abstract

Communicated by M. SapirThompson'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 .In this section we will define Thompson's group V following [6]. In that paper, a family G n,r of infinite finitely presented groups is defined, for which V = G 2,1 . We recall the basic definitions and results for the special case n = 2 and r = 1.The algebra V 2,1 Consider the free algebra V 2,1 generated by {x} with two unary operations α 1 , α 2 and a binary operation λ satisfying λ(α 1 u, α 2 u) = u for any u and α i (λ(a 1 , a 2 )) = a i .Elements in the algebra can be written as one of the following types:Fully parenthesized expressions such as ((a 1 ·a 2 )·(a 3 ·a 4 ) · · ·) where · represents the binary operation λ and each a i is of type (i). 42 O. P. Salazar-Díaz The following lemma is a particular case of [6, Lemma 4.1]. Lemma 2. If f is an element of V, then there is a unique minimal expansion W of {x} such that f (W ) ⊂ P {x}. In other words, any other expansion of {x} with this property is an expansion of W . Claim 1. The set f (W ), from the previous lemma, is an expansion Z of {x}. Proof. By [6, Lemma 2.4]. Tree pairsBy a correspondence between expansions and binary trees, elements in Thompson's group V can also be described by using trees. We will show how. Definition 3.A graph is a tree iff any two vertices are connected by a unique simple path; with a simple path being one for which no vertex appears more than once.Now, we want to construct a tree. Given two elements u, v of P {x}, define {u, v} to be an edge if u = α i v or v = α i u (i = 1, 2). Let E be the set of all edges constructed in this way. The following is standard. Claim 2. The pair T = ( P {x}, E) is a tree.Since P {x} is an infinite set, we say T is an infinite tree and we say that the root of the tree is x. If v ∈ P {x}, then v = Γx for some Γ and all Σx with Σ a suffix of Γ are vertices in the unique simple path from v to the root. If u = v is in this path we say u is above v. Note that if u 1 = u 2 are both above v then one of u 1 , u 2 is above the other.The "ends" of T are the elements of a Cantor set C. These are infinite simple paths from the root x and correspond to infinite words in the alphabet {α 1 , α 2 }.Remark. Consider a finite subset S of P {x} such that:Let E S be the set of all edges in E with both ends in S. Then T = S, E S also defines a tree. It is a subtree of T and is finite since S is finite.From now on, any tree T is a subtree (finite or infinite) of T . Definition 4.A vertex u in a tree T is a leaf if α 1 u, α 2 u are not in T .Remark. The infinite tree T does not have leaves.Definition 5. Given a tree T , a caret in T is a triple of vertices of the form (u, α 1 u, α 2 u) where all vertices in the tuple are in T .

Free products in R. Thompson’s group 𝑉

“…We will define here everything which is required for this note. Lemmas and corollaries appearing in this subsection before Lemma 2.4 can all either be found in [4] or in [12], or are easy consequences of the results found therein.…”

Computing the Scale of an Endomorphism of a totally Disconnected Locally Compact Group

“…As seen here, the scale depends on more than just the speed with which the axis is translated towards or away from the ends, but also on the "thickness" of the axis. A similar description of the dynamics of the action of almost automorphisms is given in [36], which develops ideas in [37].…”

The power conjugacy problem in Higman–Thompson groups

“…Note that other approaches to algorithmic problems in G n,r have been developed. For example [27] proposes an algorithm for the conjugacy problem in G 2,1 based on the the revealing tree pairs of Brin [8]. In [6] the same methods are used to study the centralisers of elements of G n,1 for n ≥ 2.…”