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On the outer automorphism groups of free groups

Vladimir A. Tolstykh

TL;DR

The paper proves that the outer automorphism group of a free group of countably infinite rank is complete. It develops definability results for extremal and soft involutions in $\mathrm{Out}(F)$ via canonical-basis analysis and derives a group-theoretic characterization that isolates extremal involutions. Building on the small index property and a stabilization argument, it then shows every automorphism of $\mathrm{Out}(F)$ is inner, completing the rigidity result. The work extends prior finite-rank completeness results to the infinite rank, countable case, providing a framework for potential generalization to broader infinite-rank settings.

Abstract

We prove that the outer automorphism group of a free group of countably infinite rank is complete.

On the outer automorphism groups of free groups

TL;DR

The paper proves that the outer automorphism group of a free group of countably infinite rank is complete. It develops definability results for extremal and soft involutions in via canonical-basis analysis and derives a group-theoretic characterization that isolates extremal involutions. Building on the small index property and a stabilization argument, it then shows every automorphism of is inner, completing the rigidity result. The work extends prior finite-rank completeness results to the infinite rank, countable case, providing a framework for potential generalization to broader infinite-rank settings.

Abstract

We prove that the outer automorphism group of a free group of countably infinite rank is complete.
Paper Structure (3 sections, 12 theorems, 57 equations)

This paper contains 3 sections, 12 theorems, 57 equations.

Table of Contents

  1. Involutions
  2. Extremal involutions
  3. Definability of finitary automorphisms

Key Result

Lemma 1.1

Let $\varphi$ be a soft involution of $\mathrm{Aut}(F),$ let $\mathscr B$ be a canonical basis for $\varphi$ and let $X=X(\mathscr B)$ be the set of elements of $\mathscr B$ that are inverted by $\varphi.$ (i) Suppose that an involution $g \in \mathrm{Out}(F)$ is a conjugate of $\widehat{\varphi}.$ for a suitable $x \in X;$ (iii) if for all $x \in X,$ the involutions $\varphi,\tau_x \varphi$ are

Theorems & Definitions (26)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Lemma 1.3
  • Claim 1.4
  • proof
  • Lemma 2.1
  • proof
  • Claim 2.2
  • ...and 16 more