|
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.
|
