Yet another proof of Artin's theorem (original) (raw)

On Finitely Generated Subgroups of Free Products

Journal of the Australian Mathematical Society, 1971

If H is a subgroup of a group G we shall say that G is H-residually finite if for every element g in G, outside H, there is a subgroup of finite index in G, containing H and still avoiding g. (Then, according to the usual definition, G is residually finite if it is E-residually finite, where E is the identity subgroup). Definitions of other terms used below may be found in § 2 or in [6].

On commutator equalities and stabilizers in free groups

Canadian Mathematical Bulletin, 1976

A simple proof is given of a result of Hmelevskiï on the solutions of the equation [x, y] = [u, u] over a free group for any specified u, v. To illustrate, the equation is solved explicitly for (w, v) = (a, b), (a 2 , b), ([a, b], c) (where a, b, c freely generate the free group) and thence stabilizers of the corresponding commutators in the automorphism group of this free group are determined.

On countable elementary free groups

arXiv: Logic, 2020

We prove that if a countable group is elementarily equivalent to a non-abelian free group and all of its finitely generated abelian subgroups are cyclic, then the group is a union of a chain of regular NTQ groups (i.e., hyperbolic towers).

The R∞-property for right-angled Artin groups

Topology and its Applications, 2021

Given a group G and an automorphism ϕ of G, two elements x, y ∈ G are said to be ϕ-conjugate if x = gyϕ(g) −1 for some g ∈ G. The number of equivalence classes is the Reidemeister number R(ϕ) of ϕ, and if R(ϕ) = ∞ for all automorphisms of G, then G is said to have the R ∞-property. A finite simple graph Γ gives rise to the right-angled Artin group A Γ , which has as generators the vertices of Γ and as relations vw = wv if and only if v and w are joined by an edge in Γ. We conjecture that all non-abelian right-angled Artin groups have the R ∞-property and prove this conjecture for several subclasses of right-angled Artin groups. 1 Twisted conjugacy and Reidemeister numbers Let G be a group and ϕ : G → G be an automorphism. For x, y ∈ G, we say that x and y are ϕ-conjugate and write x ∼ ϕ y if there exists a g ∈ G such that x = gyϕ(g) −1. The equivalence class of x is denoted by [x], or [x] ϕ for clarity if there are multiple automorphisms involved. We define R[ϕ] to be the set of all ∼ ϕ-equivalence classes and the Reidemeister number R(ϕ) of ϕ as the cardinality of R[ϕ]. Note that R(ϕ) ∈ N 0 ∪ {∞}. Finally, we define the Reidemeister spectrum to be Spec R (G) := {R(ϕ) | ϕ ∈ Aut(G)}. We say that G has the R ∞property, also denoted as G ∈ R ∞ , if Spec R (G) = {∞}. We say that G has full Reidemeister spectrum if Spec R (G) = N 0 ∪ {∞}. The notion of Reidemeister number arises from Nielsen fixed-point theory, where its topological analog serves as a count of the number of fixed point classes of a continuous self-map, and is strongly related to the algebraic one defined above, see [15]. It has been proven for several (classes of) groups that they possess the R ∞-property, e.g. Baumslag-Solitar groups [7] and their generalisations [19], extensions of SL(n, Z) and GL(n, Z) by a countable abelian group [20], and Thompson's group [1]. We refer the reader to [8] for a more exhaustive list of groups having the R ∞-property. In this article, we study the Reidemeister spectrum of right-angled Artin groups, RAAGs for short. Given a graph Γ with the set of vertices V , the RAAG associated to it is the group A Γ = V | [v, w] if v, w ∈ V are joined by an edge in Γ. Extreme cases of RAAGs include free groups and free abelian groups, coming from edgeless and complete graphs, respectively. From [6, Theorem 3] (see also [4]), it readily follows that all non-abelian free groups of finite rank have the R ∞-property. On the other hand, Spec R (Z) = {2, ∞} and Spec R (Z n) = N 0 ∪ {∞} for n ≥ 2 (see e.g. [22]). For groups closely 1 Research supported by long term structural funding-Methusalem grant of the Flemish Government. 2 Researcher funded by FWO PhD-fellowship fundamental research (file number: 1112520N).

Some Applications of Free Group

Journal of Multidisciplinary Modeling and Optimization, 2020

In this paper, we study many concepts as applications of free group, for example, presentation, rank of free group, and inverse of free group. We discussed some results about presentation concept and related it with free group. The our main result about free rank, is if G is a group, then G is free rank n if and only if G≅Zn. Also we obtained a new fact about inverse semigroup which say there is no free inverse semigroup is finitely generated as a semigroup. Moreover, we studied some results of inverse of free semigroup, These were illustrated by formulating Theorems, Lemma, Corollaries, and all of these concepts were explained through detailed examples.