There is a paper of Martin Lustig on his webpage giving a positive answer to the conjugacy problem for the outer automorphism group of the free group $F_n$. On the other hand, there seems not to be a publication in a journal about this, but several related publications. Is it possible to say which are the "best" results in this direction, with a complete proof ? (For example, a result by Sela if the outer automorphism group is finite, etc.).
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asked Dec 3, 2013 at 16:39
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$\begingroup$ This was listed as an open question as of 2006 (Question 27): math.cornell.edu/~vogtmann/papers/AutQuestions $\endgroup$– Ian AgolCommented Dec 4, 2013 at 5:41
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$\begingroup$ @IanAgol: yes, and I wondered what exactly is known, and has been done. Bridson and Vogtmann only mention Lustig's work, and references for the case $GL(n,\mathbb{Z})$ and the mapping class group case. $\endgroup$– Dietrich BurdeCommented Dec 4, 2013 at 8:38
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4 Answers
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One result in this direction is given by Dahmani . His algorithm will determine conjugacy for pairs of atoroidal outer automorphisms, ie automorphisms that do not fix a non-trivial conjugacy class. Note that he uses some very heavy machinery---his solution goes via the isomorphism problem for hyperbolic groups (solved by him and Guirardel along the lines pioneered by Sela).
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The paper
D.G. Kharamtsov: Solvability of the conjugacy problem for finite subgroups in automorphism groups of free groups, Algebra and Logic 34(5), 1995, 311-337
solves the conjugacy problem for finite subgroups of ${\rm Aut}(G)$ and ${\rm Out}(G)$, but states that the conjugacy problem for elements of infinite order in these groups is still open, and that the "solution appears to be a goal not easy to be reached". -- I don't know whether this information is still up-to-date.
There does not seem to be an arXiv- or even published version of Martin Lustig's preprint Structure and conjugacy for automorphisms of free groups.
answered Dec 3, 2013 at 17:40
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The solution for "general Dehn twist" automorphisms of $Out(F_n)$, namely those which are products of powers of a commuting set of individual Dehn twists, is given in
MR1691946 (2000c:20058) Cohen, Marshall M.; Lustig, Martin. The conjugacy problem for Dehn twist automorphisms of free groups. Comment. Math. Helv. 74 (1999), no. 2, 179–200.
For fully irreducible outer automorphisms, namely those for which no proper nontrivial free factor conjugacy class is periodic, although there is no publication that contains a complete description of the conjugacy problem, nonetheless it is a "folk theorem" [CORRECTION: see the post of User40911 for an attribution of this to J. Los, as I ought to have remembered since I wrote the MathSciNet review :-/ ] that this case is completely covered by the train track technology given in
MR1147956 (92m:20017 Bestvina, Mladen; Handel, Michael. Train tracks and automorphisms of free groups. Ann. of Math. (2) 135 (1992), no. 1, 1–51.
That paper is not written with algorithmic issues in mind, so there are plenty of algorithmic details to fill in. But in outline, given a fully irreducible $\phi \in Out(F_n)$: it is understood that a train track representative of $\phi$ may be computed by following the procedure described in this paper (and there exist several computer implementations); and it is also understood that all of the finitely many train track representatives of $\phi$ (up to topological conjugacy) can similarly be computed; and, finally, this finite set of data is a complete conjugacy invariant for $\phi$.
answered Dec 6, 2013 at 18:52
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Theorem 1 in http://www.zbmath.org/?q=an:00912201 (J. Loss On the conjugacy problem for automorphisms of free groups. Topology 35, n.3 p779-806, 1996) states
"The conjugacy problem for the irreducible outer automorphisms of a free group admits a solution."
