An Equivariant Whitehead Algorithm and Conjugacy for Roots of Dehn Twists

§0. Introduction In 1912 Max Dehn formulated fundamental problems concerning a group given by generators and relations. One of these, the conjugacy problem, asks whether there is an algorithm to decide whether two words in the generators represent conjugate elements of the group. Dehn himself gave an elegant solution to this problem in the case when the group is the fundamental group of a closed hyperbolic surface, given by the standard presentation. We consider aspects of the conjugacy problem for the group Out(Fn) of outer automorphisms of a free group of rank n. For finite-order automorphisms, and in fact for finite groups of automorphisms, an algorithm to solve the conjugacy problem follows from results of Krstić [7]. For Dehn twist automorphisms, which are automorphisms given in terms of a graph-of-groups decomposition of Fn by specifying a twistor in every edge group, an algorithm to solve the conjugacy problem was given by Cohen and Lustig [4]. In the present paper, we start with the conjugacy problem for finite groups of automorphisms, with the additional constraint that the conjugating automorphism must take one given finite set of words to another. In Section 1 we describe an equivariant Whitehead algorithm and prove: