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

Armando Martino

Algebraic & Geometric Topology 2 (2002) 897–919

arXiv: math.GR/0101130


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
Mathematical Subject Classification 2000
Primary: 20E05, 20E36
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Received: 4 February 2002
Accepted: 21 August 2002
Published: 20 October 2002
Armando Martino
Department of Mathematics
University College Cork