  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.

Group Theory