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The conjugacy problem for Dehn twist automorphisms of free groups. (English) Zbl 0956.20021
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 S. Krstic, M. Lustig and 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 M. Lustig’s survey preprint 1994 [Prime factorization and conjugacy problem in \(\text{Out}(F_n)\)].

MSC:
20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F28 Automorphism groups of groups
57M07 Topological methods in group theory
20E08 Groups acting on trees
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