Orbit-blocking words in free groups

TL;DR

This paper strengthens existing results on primitivity-blocking words in free groups, showing that certain words cannot appear as subwords in any automorphic image of a given element, impacting Whitehead's problem complexity.

Contribution

It introduces a new class of words that cannot be subwords of automorphic images, advancing understanding of free group automorphisms and complexity.

Findings

Existence of words that cannot be subwords of any automorphic image of a given element.

Implications for the average-case complexity of Whitehead's problem.

Enhanced results on primitivity-blocking words in free groups.

Abstract

By strengthening known results about primitivity-blocking words in free groups, we prove that for any nontrivial element w of a free group of finite rank, there are words that cannot be subwords of any cyclically reduced automorphic image of w. This has implications for the average-case complexity of a variant of Whitehead's problem.