Classification of automorphic conjugacy classes in the free group on two generators

TL;DR

This paper provides a complete classification of automorphic conjugacy classes in the free group on two generators by associating finite directed graphs, refining previous work and establishing bounds on Whitehead's algorithm efficiency.

Contribution

It introduces a structural classification of automorphic conjugacy classes in F_2 through graph association, confirming a conjecture and improving algorithmic bounds.

Findings

Complete classification of automorphic conjugacy classes in F_2

Refinement of previous classifications by Khan

Sharp upper bound on Whitehead's algorithm runtime

Abstract

We associate a finite directed graph with each equivalence class of words in $F_2$ under $\operatorname*{Aut} F_2$, and we completely classify these graphs, giving a structural classification of the automorphic conjugacy classes of $F_2$. This classification refines work of Khan and proves a conjecture of Myasnikov and Shpilrain on the number of minimal words in an automorphic conjugacy class whose minimal words have length $n$, which in turn implies a sharp upper bound on the running time of Whitehead's algorithm for determining whether two words in $F_2$ are automorphic conjugates.