Clusters, Currents, and Whitehead's Algorithm

Abstract

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.