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An algorithm to decide if an outer automorphism is geometric

Edgar A. Bering, Yulan Qing, Derrick R. Wigglesworth

Abstract

An outer automorphism of a free group is geometric if it can be represented by a homeomorphism of a compact surface. Bestvina and Handel gave an algorithmic characterization of geometric irreducible outer automorphisms using relative train tracks in 1995. The general case of detecting geometric outer automorphisms remained open, with a few partial results appearing subsequently. In this paper we give a complete resolution to the problem: an algorithm that can decide if a general outer automorphism is geometric. The algorithm is constructive and produces a realizing surface homeomorphism if one exists. We make use of advances in train-track theory, in conjunction with the Guirardel core of tree actions and Nielsen-Thurston theory for surfaces.

An algorithm to decide if an outer automorphism is geometric

Abstract

An outer automorphism of a free group is geometric if it can be represented by a homeomorphism of a compact surface. Bestvina and Handel gave an algorithmic characterization of geometric irreducible outer automorphisms using relative train tracks in 1995. The general case of detecting geometric outer automorphisms remained open, with a few partial results appearing subsequently. In this paper we give a complete resolution to the problem: an algorithm that can decide if a general outer automorphism is geometric. The algorithm is constructive and produces a realizing surface homeomorphism if one exists. We make use of advances in train-track theory, in conjunction with the Guirardel core of tree actions and Nielsen-Thurston theory for surfaces.
Paper Structure (29 sections, 40 theorems, 41 equations, 3 figures, 2 algorithms)

This paper contains 29 sections, 40 theorems, 41 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs, paths, and splittings
  4. Lines and laminations
  5. Free factor systems and supports
  6. $\mathop{\mathrm{Out}}\nolimits({\mathbb F})$ and surface homeomorphisms
  7. Automorphisms and lifts
  8. Principal lifts and rotationless outer automorphisms
  9. Principal lifts and the Nielsen approach to $\mathop{\mathrm{Map}}\nolimits(\Sigma)$
  10. Train tracks, splittings, and CTs
  11. Computing with CTs
  12. The (equivariant) Whitehead algorithm
  13. Guirardel's core
  14. Geometric models of geometric EG strata
  15. Defining the models
  16. ...and 14 more sections

Key Result

Theorem A

Let $\phi \in \mathop{\mathrm{Out}}\nolimits({\mathbb F})$ be an outer automorphism. There exists an algorithm that decides if $\phi$ is geometric and a procedure to compute a realization of $\phi$ as a surface homeomorphism if one exists.

Figures (3)

  • Figure 1: A surface homeomorphism $g\colon \Sigma\to\Sigma$ with associated CT $f\colon G\to G$.
  • Figure 2: $\mathop{\mathrm{Core}}\nolimits(T, T\phi)/{\mathbb F}$ for a typical Dehn twist.
  • Figure 3: The surface $S$ in which the Guirardel core embeds.

Theorems & Definitions (92)

  • Theorem A
  • Corollary B
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Dehn-Nielsen-Baer Theorem
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5: Thurston Normal Form FLPWu:NF
  • Definition 2.6
  • ...and 82 more