#### 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.

Notes on file formats
Armando Martino

Department of Mathematics, University College Cork

Cork, Ireland

Email: A.Martino@ucc.ie

AGT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**