Automorphic orbits in Free Groups: Words versus Subgroups

We show that the following problems are decidable in rank 2 free groups: does a given finitely generated subgroup H contain primitive elements? and does H meet the automorphic orbit of a given word u? 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.

• Type: Journal
• Year: 2010
• Where: IJAC
• Authors: Pedro V. Silva, Pascal Weil
• Keywords: Automorphic Orbit, Automorphism, IJAC 2010, Primitive Elements