Abstract
Abstract Let F 2 F_{2} denote the free group of rank 2. Our main technical result of independent interest is the following: for any element 𝑢 of F 2 F_{2} , there is some g ∈ F 2 g\in F_{2} such that no cyclically reduced image of 𝑢 under an automorphism of F 2 F_{2} contains 𝑔 as a subword. We then address the computational complexity of the following version of the Whitehead automorphism problem: given a fixed u ∈ F 2 u\in F_{2} , decide, on an input v ∈ F 2 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.
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