2 00 9 Automorphic orbits in free groups : words versus subgroups ∗

We show that the following problems are decidable in a rank 2 free group F2: does a given finitely generated subgroupH contain primitive elements? and does H meet the orbit of a given word u under the action of G, the group of automorphisms of F2? 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, the following weaker problem is decidable: 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. 2000 Mathematics Subject Classification: 20E05 The first author acknowledges support from Project ASA (PTDC/MAT/65481/2006) and C.M.U.P., financed by F.C.T. (Portugal) through the programmes POCTI and POSI, with national and European Community structural funds. Both authors acknowledge support from the ESF project AutoMathA. Centro de Matemática, Faculdade de Ciências, Universidade do Porto, R. Campo Alegre 687, 4169-007 Porto, Portugal LaBRI, Université Bordeaux-1, 351 cours de la Libération, 33405 Talence Cedex, France