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

Armando Martino

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.

Keywords. Free group, automorphism

AMS subject classification. Primary: 20E05, 20E36.

DOI: 10.2140/agt.2002.2.897

E-print: arXiv:math.GR/0101130

Submitted: 4 February 2002. Accepted: 21 August 2002. Published: 20 October 2002.

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Armando Martino
Department of Mathematics, University College Cork
Cork, Ireland

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