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In this paper we define a quantity called the rank of an outer automorphismof 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 forautomorphisms of free groups', Duke Math. J. 93 (1998) 425-452] without thecount of fixed points on the boundary. We proceed to analyze outerautomorphisms 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 withmaximal rank fixed point subgroup fixes a primitive element', J. Pure andApplied Algebra 88 (1993) 43-49]. We also deduce that all such outerautomorphisms can be represented by Dehn twists, thus proving the converse to aresult in [M.M. Cohen and M. Lustig, `The conjugacy problem for Dehn twistautomorphisms of free groups', Comment Math. Helv. 74 (1999) 179-200], andindicate a solution to the conjugacy problem when such automorphisms are givenin terms of images of a basis, thus providing a moderate extension to the maintheorem of Cohen and Lustig by somewhat different methods.

Maximal index automorphisms of free groups with no attracting fixed points on the boundary are Dehn twists