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AN EQUIVARIANT WHITEHEAD ALGORITHM AND CONJUGACY FOR ROOTS OF DEHN TWIST AUTOMORPHISMS

Published online by Cambridge University Press:  20 January 2009

Sava Krstić ,
Martin Lustig  and
Karen Vogtmann
Sava Krstić
Affiliation:
Department of Computer Science and Engineering, Oregon Graduate Institute, PO Box 91000, Portland, OR 97291, US (krstic@cse.ogi.edu)
Martin Lustig
Affiliation:
Laboratoire de Mathématiques, Université d’Aix–Marseille III, Ave. E. Normandie-Niemen, 13397 Marseille 20, FR (Martin.Lustig@math.u-3mrs.fr)
Karen Vogtmann
Affiliation:
Department of Mathematics, 555 Malott Hall, Cornell University, Ithaca, NY 14853, US (vogtmann@math.cornell.edu)
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Abstract

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Given finite sets of cyclic words $\{u_1,\dots,u_k\}$ and $\{v_1,\dots,v_k\}$ in a finitely generated free group $F$ and two finite groups $A$ and $B$ of outer automorphisms of $F$, we produce an algorithm to decide whether there is an automorphism which conjugates $A$ to $B$ and takes $u_i$ to $v_i$ for each $i$. If $A$ and $B$ are trivial, this is the classic algorithm due to Whitehead. We use this algorithm together with Cohen and Lustig’s solution to the conjugacy problem for Dehn twist automorphisms of $F$ to solve the conjugacy problem for outer automorphisms which have a power which is a Dehn twist. This settles the conjugacy problem for all automorphisms of $F$ which have linear growth.

AMS 2000 Mathematics subject classification: Primary 20F32. Secondary 57M07

Keywords

automorphisms of free groupsconjugacy problemDehn twistsWhitehead algorithm
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 44 , Issue 1 , February 2001 , pp. 117 - 141
Copyright
Copyright © Edinburgh Mathematical Society 2001