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

Given finite sets of cyclic words {u(1), ..., u(k)} and {nu (1), ..., nu (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 nu (i) for each i. If 4 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 serve 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.