An equivariant Whitehead algorithm and conjugacy for roots of Dehn twists

Sava Krstic, Martin Lustig and Karen Vogtmann

Proceedings of the Edinburgh Mathematical Society (2001) 44, 117-141.

Given finite sets of cyclic words {u₁,...,u_k} and {v₁,...,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 _i for each i. If A and B are trivial, this is the classic algorithm due to J.H.C. 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.

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