An algorithm to decide if an outer automorphism is geometric

TL;DR

This paper presents a complete algorithmic solution to determine whether a given outer automorphism of a free group is geometric, building on advances in train-track theory, Guirardel core, and Nielsen-Thurston theory.

Contribution

It provides the first complete, constructive algorithm to decide if an outer automorphism is geometric, extending previous partial results.

Findings

Algorithm successfully decides geometric outer automorphisms

Produces a surface homeomorphism when one exists

Integrates train-track theory, Guirardel core, and Nielsen-Thurston theory

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.