We solve the twisted conjugacy problem on Thompson's group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut(+)(F) are orbit decidable provided a certain conjecture on Thompson's group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak-Fel'shtyn-Gon double dagger alves in [4] showing that F has property R (a), and which can be extended to show that Thompson's group T also has property R (a).
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机译：我们解决了汤普森群F上的扭曲共轭问题。我们还展示了Aut（F）的轨道不可确定的子群，并给出了Aut（F）和Aut（+）（F）是轨道可确定的证明，这在Thompson的群上有一定的猜想T是真的。通过使用Bogopolski，Martino和Ventura在[5]中引入的一般标准，我们构建了F的自由扩展族，其中F的共轭问题无法解决。作为我们技术的副产品，我们在[4]中给出了Bleak-Fel'shtyn-Gon双匕首alves的结果的新证明，该结果表明F具有属性R（a），并且可以扩展为表明Thompson's group T也具有属性R（a）。
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