Clusters, currents and Whitehead's algorithm

Abstract

Using geodesic currents, we provide a theoretical justification for some of the experimental results regarding the behavior of Whitehead's algorithm on non-minimal inputs, that were obtained by Haralick, Miasnikov and Myasnikov via pattern recognition methods. In particular we prove that the images of "random" elements of a free group $F$ under the automorphisms of $F$ form "clusters" that share similar normalized Whitehead graphs and similar behavior with respect to Whitehead's algorithm.