  In this paper we define a quantity called the rank of an outer automorphism
of a free group which is the same as the index introduced in [D. Gaboriau, A.
Jaeger, G. Levitt and M. Lustig, `An index for counting fixed points for
automorphisms of free groups', Duke Math. J. 93 (1998) 425-452] without the
count of fixed points on the boundary. We proceed to analyze outer
automorphisms of maximal rank and obtain results analogous to those in [D.J.
Collins and E. Turner, `An automorphism of a free group of finite rank with
maximal rank fixed point subgroup fixes a primitive element', J. Pure and
Applied Algebra 88 (1993) 43-49]. We also deduce that all such outer
automorphisms can be represented by Dehn twists, thus proving the converse to a
result in [M.M. Cohen and M. Lustig, `The conjugacy problem for Dehn twist
automorphisms of free groups', Comment Math. Helv. 74 (1999) 179-200], and
indicate a solution to the conjugacy problem when such automorphisms are given
in terms of images of a basis, thus providing a moderate extension to the main
theorem of Cohen and Lustig by somewhat different methods.

Group Theory