Displacements of automorphisms of free groups II: Connectivity of level sets and decision problems

TL;DR

This paper proves that all level sets of displacement functions for automorphisms of free groups are connected, aiding in algorithmic solutions to conjugacy and irreducibility detection problems.

Contribution

It establishes the connectivity of level sets of displacement functions for all automorphisms of free groups, extending previous results to reducible cases and general deformation spaces.

Findings

Connected level sets enable stopping procedures in algorithms.

Reproves conjugacy problem for irreducible automorphisms.

Provides a method to detect automorphism irreducibility.

Abstract

This is the second of two papers in which we investigate the properties of displacement functions of automorphisms of free groups (more generally, free products) on the Culler-Vogtmann Outer space $CV_n$ and its simplicial bordification. We develop a theory for both reducible and irreducible autormorphisms. As we reach the bordification of $CV_n$ we have to deal with general deformation spaces, for this reason we developed the theory in such generality. In first paper~\cite{FMpartI} we studied general properties of the displacement functions, such as well-orderability of the spectrum and the topological characterization of min-points via partial train tracks (possibly at infinity). This paper is devoted to proving that for any automorphism (reducible or not) any level set of the displacement function is connected. As an application, this result provides a stopping procedure for brute force search algorithms in $CV_n$. We use this to reprove two known algorithmic results: the conjugacy problem for irreducible automorphisms and detecting irreducibility of automorphisms. Note: the two papers were originally packed together in the preprint arxiv:1703.09945. We decided to split that paper following the recommendations of a referee.