An equivariant Whitehead algorithm and conjugacy for roots of Dehn twist automorphisms (Q2716983)

The paper is a step towards a complete solution of the conjugacy problem for the outer automorphism group \(\text{Out}(F)\) of a free group \(F\) of finite rank: does there exist an algorithm to decide whether two given outer automorphisms of \(F\) are conjugate or not? This problem has been solved for various types of outer automorphisms: for finite order elements (and more generally for finite subgroups) of \(\text{Out}(F)\), for irreducible automorphisms, for Dehn twist automorphisms (which include and generalize the automorphisms induced by Dehn twists of surfaces).NEWLINENEWLINENEWLINEIn the present paper, using the solution of the conjugacy problem for Dehn twist automorphisms due to \textit{M. M. Cohen} and \textit{M. Lustig} [Comment. Math. Helv. 74, No. 2, 179-200 (1999; Zbl 0956.20021)], the conjugacy problem is solved for outer automorphisms which have powers that are Dehn twist automorphisms; these are exactly the automorphisms which have linear growth. To a Dehn twist automorphism is associated an invariant decomposition of \(F\) as a graph of groups, and a finite root of a Dehn twist automorphism permutes resp. induces finite order automorphisms of the vertex groups. It is shown that, up to some combinatorial data, two roots of Dehn twists automorphisms are conjugate if and only if these finite-order automorphisms of the vertex groups are conjugate by an automorphism preserving the edge groups of adjacent edges. This can be decided by the following equivariant Whitehead algorithm which is the first main result of the paper. Given two finite subgroups \(A\) and \(B\) of \(\text{Out}(F)\) and two finite sets of cyclic words \(u_i\) and \(v_i\) in \(F\), there exists an explicit algorithm which decides whether \(A\) and \(B\) are conjugate by an automorphism of \(F\) which takes \(u_i\) to \(v_i\), for each \(i\). If \(A\) and \(B\) are trivial this is the classical Whitehead algorithm, without any \(u_i\) and \(v_i\) it is the above-mentioned solution of the conjugacy problem for finite subgroups of \(\text{Out}(F_n)\) (following from work of Krstić). The main technical tool is an equivariant peak reduction lemma for equivariant Whitehead moves between \(G\)-graphs with a finite invariant set of homotopy classes of loops, for a finite group \(G\).