The conjugacy problem for Dehn twist automorphisms of free groups
A Dehn twist \(D=D({\mathcal G},(z_e)_{e\in E}({\mathcal G}))\) consists of a graph of groups \(\mathcal G\) and for every edge \(e\) of \(\mathcal G\) a specified \(z_e\) in the centre of the edge group \(G_e\). This determines a Dehn twist automorphism \(D_\upsilon\) of the fundamental group \(\pi_1({\mathcal G},\upsilon)\) for each vertex \(\upsilon\) of \(\mathcal G\) and hence an automorphism of the abstract group \(\pi_1({\mathcal G})\) which is well defined up to inner automorphisms. The classic example is that of an automorphism of the fundamental group of a surface which is induced by a Dehn twist homeomorphism of the surface. Thus \(D\) determines an outer automorphism \(\widehat D\in\text{Out}(\pi({\mathcal G}))\). A Dehn twist automorphism of the free group \(F_n\) is an automorphism which is conjugate to such a Dehn twist automorphism of \(\pi_1({\mathcal G},\upsilon)\) for some graph of groups \(\mathcal G\). The main result is an algorithm which decides whether two given Dehn twist automorphisms of the free group \(F_n\) are conjugate or conjugate up to an inner automorphism. It is also announced that the results of the present paper will be extended in the forthcoming paper by \textit{S. Krstic}, \textit{M. Lustig} and \textit{K. Vogtmann} [An equivariant Whitehead algorithm and conjugacy for roots of Dehn twists (preprint 1997)] to roots of Dehn twist automorphisms (and hence to all automorphisms of \(F_n\) of linear growth) and will be a key part of the second author's complete solution of the conjugacy problem for \(\text{Out}(F_n)\) as announced in \textit{M. Lustig}'s survey preprint 1994 [Prime factorization and conjugacy problem in \(\text{Out}(F_n)\)].
scientific article; zbMATH DE number 1333319