Mathematics – Group Theory
Scientific paper
2001-01-16
Algebr. Geom. Topol. 2 (2002) 897-919
Mathematics
Group Theory
Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-36.abs.html
Scientific paper
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.
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