Detecting fully irreducible automorphisms: a polynomial time algorithm. With an appendix by Mark C. Bell

TL;DR

This paper presents a polynomial time algorithm for determining whether an automorphism of a free group is fully irreducible, refining previous methods by removing inefficiencies and broadening applicability.

Contribution

The authors develop a more efficient, polynomial time algorithm for detecting fully irreducible automorphisms in Out(F_N), including a new train track criterion and an alternative Nielsen path algorithm.

Findings

Polynomial time decision algorithm for fully irreducible automorphisms.

A train track criterion covering all fully irreducible elements.

An efficient algorithm for finding indivisible Nielsen paths.

Abstract

In \cite{Ka14} we produced an algorithm for deciding whether or not an element $\phi\in Out(F_N)$ is an iwip ("fully irreducible") automorphism. At several points that algorithm was rather inefficient as it involved some general enumeration procedures as well as running several abstract processes in parallel. In this paper we refine the algorithm from \cite{Ka14} by eliminating these inefficient features, and also by eliminating any use of mapping class groups algorithms. Our main result is to produce, for any fixed $N\ge 3$, an algorithm which, given a topological representative $f$ of an element $\phi$ of $Out(F_N)$, decides in polynomial time in terms of the "size" of $f$, whether or not $\phi$ is fully irreducible. In addition, we provide a train track criterion of being fully irreducible which covers all fully irreducible elements of $Out(F_N)$, including both atoroidal and non-atoroidal ones. We also give an algorithm, alternative to that of Turner, for finding all the indivisible Nielsen paths in an expanding train track map, and estimate the complexity of this algorithm. An appendix by Mark Bell provides a polynomial upper bound, in terms of the size of the topological representative, on the complexity of the Bestvina-Handel algorithm \cite{BH92} for finding either an irreducible train track representative or a topological reduction.