DISPLACEMENTS OF AUTOMORPHISMS OF FREE GROUPS II: CONNECTIVITY OF LEVEL SETS AND DECISION PROBLEMS

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 CVn and its simplicial bordification. We develop a theory for both reducible and irreducible autormorphisms. As we reach the bordification of CVn we have to deal with general deformation spaces, for this reason we developed the theory in such generality. In our previous first paper 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. Here, by the “level set” we intend to indicate the set of points displaced by at most some amount, rather than exactly some amount; this is sometimes called a “sub-level set”. As an application, this result provides a stopping procedure for brute force search algorithms in CVn. We use this to reprove two known algorithmic results: the conjugacy problem for irreducible automorphisms and detecting irreducibility of automorphisms.