Parageometric Outer Automorphisms of Free Groups

We study those fully irreducible outer automorphisms φ of a finite rank free group Fr which are parageometric, meaning that the attracting fixed point of φ in the boundary of outer space is a geometric R-tree with respect to the action of Fr, but φ itself is not a geometric outer automorphism in that it is not represented by a homeomorphism of a surface. Our main result shows that the expansion factor of φ is strictly larger than the expansion factor of φ−1. As corollaries (proved independently by Guirardel), the inverse of a parageometric outer automorphism is neither geometric nor parageometric, and a fully irreducible outer automorphism φ is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric