Clusters, currents, and whitehead’s algorithm

Using geodesic currents, we provide a theoretical justification for some of the experimental results obtained by Haralick, Miasnikov, and Myasnikov via pattern-recognition methods regarding the behavior of Whitehead’s algorithm on nonminimal inputs. 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.