  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.
