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

Abstract

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.