A recursive presentation for Mihailova’s subgroup

Abstract

Abstract. An explicit recursive presentation for Mihailova's subgroup M.H / of F n F n corresponding to a finite, concise and Peiffer aspherical presentation H D hx 1 ; : : : ; x n j R 1 ; : : : ; R m i is given. This partially answers a question of R. I. Grigorchuk. As a corollary, we construct a finitely generated recursively presented orbit undecidable subgroup of Aut.F 3 /.Mathematics Subject Classification (2010). 20F05, 20F10.

Groups With Two Generators Having Unsolvable Word Problem and Presentations of Mihailova Subgroups of Braid Groups

“…However, Bogopolski and Venturawe [5] have given an explicit countable presentation with finite number of generators and countably infinite number of relators for Mihailova subgroup M H of F k × F k (k ≥ 2) provided that the group H can be defined by a finite, concise, and Peiffer aspherical presentation as in the following theorem. x k , and let H = x 1…”

“…One can refer to [5] for the definition of being Peiffer aspherical. Theorems 3.1 and 4.2 and Lemma 5.1 in [10] imply that, respectively, the Peiffer asphericity is preserved under HNN-extensions (Higman, Neumann, and Neumann's extension, also see [10]), under free products, and under Tietze transformations.…”

HNN extensions and stackable groups

“…Let A be the Mihailova subgroup [17] showed that the subgroup membership problem for A in H is not decidable; that is, there does not exist an algorithm that upon input of a word w in the generating set (Y ∪ Y ′ ) ±1 of H, can determine whether w represents an element of the subgroup A. The group A is finitely generated (see for example the paper of Bogopolski and Ventura [4] for a discussion and recursive presentation for this group); let Z be a finite generating set for A.…”

A Higman embedding preserving asphericity

“…A group presentation P is called (topologically) aspherical if the universal cover of the presentation complex 3 of P is contractible. The presentation P is combinatorially aspherical [6,Section 6] (sometimes also called Peiffer aspherical [4]) if every spherical van Kampen diagram over P (i.e. a van Kampen diagram on a disc with empty boundary) is (combinatorially) homotopic to a diagram without cells [5], [6].…”

“…). 4 Let H(A) be the group given by the presentation The words β(a, x), β (a, x), χ(a, x), χ (a, x), φ(a, y), ψ(a, y), γ(b, y) are called the large sections of the defining relators and the corresponding sections of boundaries of cells in the van Kampen diagrams will be called large sections of the cells. The following lemma immediately follows from the fact that the set of all large sections of defining relators satisfies the property C ( 1 12 ).…”