We investigate free product structures in R. Thompson's group V , primarily by studying the topological dynamics associated with V 's action on the Cantor Set. We show that the class of free products which can be embedded into V includes the free product of any two finite groups, the free product of any finite group with Q/Z, and the countable non-abelian free groups. We also show the somewhat surprising result that Z 2 * Z does not embed in V , even though V has many embedded copies of Z 2 and has many embedded copies of free products of pairs of its subgroups.Question 1. Is it true that if G and H are non-trivial subgroups of V , with | G |≥ 3 and G, H ∼ = G * H in V , then there are distinct non-empty sets P G and P H in C so that for any non-trivial elements g ∈ G and h ∈ H we have P H g ⊂ P G and P G h ⊂ P H ?Colloquially, must every free product of groups in V arise from a Ping-Pong in V ? Let CF PV be the smallest class of groups which contains B and which is closed under 1. isomorphism, 2. passing to subgroups,
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 .
The concept of generalized digroup was proposed by Salazar-Díaz, Velásquez and Wills-Toro in their paper “Generalized digroups” as a non trivial extension of groups. In this way, many concepts and results given in the category of groups can be extended in a natural form to the category of generalized digroups. The aim of this paper is to present the construction of the free generalized digroup and study its properties. Although this construction is vastly different from the one given for the case of groups, we will use this concept, the classical construction for groups and the semidirect product to construct the tensor generalized digroup as well as the semidirect product of generalized digroups. Additionally, we give a new structural result for generalized digroups using compatible actions of groups and an equivariant map from a group set to the group corresponding to notions of associative dialgebras and augmented racks.
The aim of this paper is to propose two possible ways of defining a g-digroup action and a first approximation to representation theory of g-digroups.
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