Algebraic and Geometric Topology 2 (2002),
paper no. 36, pages 897-919.
Maximal index automorphisms of free groups with no attracting fixed points on the boundary are Dehn twists
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.
Free group, automorphism
AMS subject classification.
Primary: 20E05, 20E36.
Submitted: 4 February 2002.
Accepted: 21 August 2002.
Published: 20 October 2002.
Notes on file formats
Department of Mathematics, University College Cork
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