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Automorphic orbits in free groups: recent progress

Vladimir Shpilrain

TL;DR

The paper surveys recent progress on asymptotic properties of automorphic orbits in free groups, focusing on counting potentially positive elements and the complexity of Whitehead's problem. It develops orbit-blocking and primitivity-blocking frameworks to study which words can appear in automorphic images, outlines generic- and average-case analyses for Whitehead's algorithm, and discusses the structure and growth of potentially positive elements and translation-equivalence phenomena. The main contributions include explicit constructions of $w$-orbit-blocking words from primitivity-blocking words, polynomial and average-case bounds for the Whitehead problem in low ranks, and the identification of open problems such as the growth rate $P(r,n)$ of potentially positive words and the exact behavior of translation equivalence in higher ranks. The significance lies in clarifying the computational landscape of automorphic imagery in $F_r$, with implications for decision problems in free groups and related areas of geometric group theory.

Abstract

In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially positive elements of a given length. We also discuss complexity (worst-case, average-case, and generic-case) of Whitehead's automorphism problem and relevant properties of automorphic orbits, including orbit-blocking words.

Automorphic orbits in free groups: recent progress

TL;DR

The paper surveys recent progress on asymptotic properties of automorphic orbits in free groups, focusing on counting potentially positive elements and the complexity of Whitehead's problem. It develops orbit-blocking and primitivity-blocking frameworks to study which words can appear in automorphic images, outlines generic- and average-case analyses for Whitehead's algorithm, and discusses the structure and growth of potentially positive elements and translation-equivalence phenomena. The main contributions include explicit constructions of -orbit-blocking words from primitivity-blocking words, polynomial and average-case bounds for the Whitehead problem in low ranks, and the identification of open problems such as the growth rate of potentially positive words and the exact behavior of translation equivalence in higher ranks. The significance lies in clarifying the computational landscape of automorphic imagery in , with implications for decision problems in free groups and related areas of geometric group theory.

Abstract

In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially positive elements of a given length. We also discuss complexity (worst-case, average-case, and generic-case) of Whitehead's automorphism problem and relevant properties of automorphic orbits, including orbit-blocking words.

Paper Structure

This paper contains 7 sections, 8 theorems.

Table of Contents

  1. Introduction
  2. Orbit-blocking words
  3. Primitivity-blocking words
  4. Whitehead's automorphism problem
  5. Generic-case and average-case complexity of Whitehead's problem
  6. Potentially positive elements
  7. Translation equivalence

Key Result

Theorem 1

Let $w\in F_r$ have cyclically reduced length $\ell$. Let $\{v_i\}_{i=1}^{\ell+1}$ be a sequence of primitivity-blocking words such that there is no cancellation between adjacent terms of the sequence, nor between $v_{\ell+1}$ and $v_1$. Then $\prod_{i=1}^{\ell+1}v_i$ is $w$-orbit-blocking.

Theorems & Definitions (10)

  • Theorem 1
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 2
  • Theorem 3
  • Theorem 1