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Pré-Publication, Document De Travail Année : 2009

Automorphic orbits in free groups: words versus subgroups

Résumé

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 even 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.
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Dates et versions

hal-00324544 , version 1 (25-09-2008)
hal-00324544 , version 2 (23-04-2009)
hal-00324544 , version 3 (02-04-2010)

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Pedro Silva, Pascal Weil. Automorphic orbits in free groups: words versus subgroups. 2009. ⟨hal-00324544v2⟩
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