Orbit-blocking words and the average-case complexity of Whitehead's problem in the free group of rank 2

TL;DR

This paper proves a technical result about orbit-blocking words in free groups and demonstrates that the Whitehead automorphism problem can be solved efficiently on average for fixed elements in F_2.

Contribution

It introduces a novel orbit-blocking words result and shows that the Whitehead automorphism problem has constant average-case complexity for fixed elements.

Findings

Existence of g that blocks cyclically reduced images of any element u in F_2

An algorithm with constant average-case complexity for the Whitehead automorphism problem

Improved understanding of automorphism complexity in free groups

Abstract

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.