Orbit-blocking words and the average-case complexity of Whitehead's problem in the free group of rank 2
<|reference_start|>Orbit-blocking words and the average-case complexity of Whitehead's problem in the free group of rank 2: Let F_2 denote the free group of rank 2. Our main technical result of independent interest is: for any element u of F_2, there is g in F_2 such that no cyclically reduced image of u under an automorphism of F_2 contains g as a subword. We then address computational complexity of the following version of the Whitehead automorphism problem: given a fixed u in F_2, decide, on an input v in F_2 of length n, whether or not v is an automorphic image of u. We show that there is an algorithm that solves this problem and has constant (i.e., independent of n) average-case complexity.<|reference_end|>
@article{hyde2024orbit-blocking,
    title={Orbit-blocking words and the average-case complexity of Whitehead's
  problem in the free group of rank 2},
    author={Lucy Hyde, Siobhan O'Connor, Vladimir Shpilrain},
    journal={arXiv preprint arXiv:2406.12322},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.12322},
    primaryClass={math.GR cs.CC}
}