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

TL;DR

This paper proves the connectedness of level sets of free group automorphisms in Culler-Vogtmann space, providing metric solutions to conjugacy and reducibility problems, with broader implications for deformation spaces.

Contribution

It establishes the connectedness of automorphism level sets in a general setting and offers metric solutions to key decision problems in free group automorphisms.

Findings

Level sets are connected in Culler-Vogtmann space.

Provides metric solutions to conjugacy and reducibility detection.

Displacements form a well-ordered set.

Abstract

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.