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

Lucy Hyde, Siobhan O'Connor, Vladimir Shpilrain

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.

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

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.
Paper Structure (4 sections, 5 theorems, 7 equations)

This paper contains 4 sections, 5 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Orbit-blocking words
  3. Average-case complexity of the Whitehead problem in $F_2$
  4. Acknowledgement

Key Result

Theorem 1

For any element $w$ of $F_2$, there is $g \in F_2$ such that no cyclically reduced image of $w$ under an automorphism of $F_2$ contains $g$ as a subword. Such words $g$ can be produced explicitly for any given $w$.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['Orbit']}
  • Theorem 3
  • proof