An algorithm to decide if an outer automorphism is geometric

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

Published 2023-10-06Version 1

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 geometricity for irreducible automorphisms, using relative train tracks. Using advances in train-track theory, in conjunction with the Guirardel core of tree actions and Nielsen-Thurston theory for surfaces, we give 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.