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