Automorphic orbits in free groups: words versus subgroups

TL;DR

This paper proves the decidability of certain orbit and subgroup membership problems in free groups of rank 2, extending some results to higher ranks with restrictions on automorphisms.

Contribution

It establishes decidability results for primitive element containment and orbit problems in free groups, including cases involving rational subsets and automorphism restrictions.

Findings

Decidability of primitive element containment in rank 2 free groups.

Decidability of orbit membership under automorphisms in rank 2.

Decidability of a weaker problem involving k-almost bounded automorphisms in higher rank.

Abstract

We show that the following problems are decidable in a rank 2 free group F_2: does a given finitely generated subgroup H contain primitive elements? and does H meet the orbit of a given word u under the action of G, the group of automorphisms of F_2? Moreover, decidability subsists if we allow H to be a rational subset of F_2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, we show the decidability of the following weaker problem: given a finitely generated subgroup H, a word u and an integer k, does H contain the image of u by some k-almost bounded automorphism? An automorphism is k-almost bounded if at most one of the letters has an image of length greater than k.