A recursive presentation for Mihailova's subgroup

TL;DR

This paper provides an explicit recursive presentation for Mihailova's subgroup related to certain group presentations, partially answering a question by Grigorchuk and constructing a finitely generated orbit undecidable subgroup of automorphisms of a free group.

Contribution

It introduces a new recursive presentation for Mihailova's subgroup associated with concise, Peiffer aspherical presentations, and constructs a finitely generated orbit undecidable subgroup of Aut(F_3).

Findings

Explicit recursive presentation for Mihailova's subgroup

Partial answer to Grigorchuk's problem 4.14

Construction of a finitely generated orbit undecidable subgroup

Abstract

We give an explicit recursive presentation for Mihailova's subgroup $M(H)$ of $F_n \times F_n$ corresponding to a finite, concise and Peiffer aspherical presentation $H=< x_1,..., x_n \mid R_1,..., R_m>$. This partially answers a question of R.I. Grigorchuk, [8, Problem 4.14]. As a corollary, we construct a finitely generated recursively presented orbit undecidable subgroup of $Aut(F_3)$.