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.
Keywords
Cite
@article{arxiv.2310.04402,
title = {An algorithm to decide if an outer automorphism is geometric},
author = {Edgar A. Bering and Yulan Qing and Derrick R. Wigglesworth},
journal= {arXiv preprint arXiv:2310.04402},
year = {2024}
}
Comments
38 page, 3 figures, 2 algorithm displays; updated to expand and clarify introduction and correct typos