On the connectivity of level sets of automorphisms of free groups, with applications to decision problems

We show that the level sets of automorphisms of free groups with respect to the Lipschitz metric are connected as subsets of Culler-Vogtmann space. In fact we prove our result in a more general setting of deformation spaces. As applications, we give metric solutions of the conjugacy problem for irreducible automorphisms and the detection of reducibility. We additionally prove technical results that may be of independent interest --- such as the fact that the set of displacements is well ordered.