The Conjugacy Problem in Extensions of Thompson's group F

Israel Journal of Mathematics, 216(1) (2016), 15-59
https://link.springer.com/content/pdf/10.1007%2Fs11856-016-1403-9.pdf (preprint - http://arxiv.org/abs/1307.6750)

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-Goncalves in [4] showing that F has property R_\infty, and which can be extended to show that Thompson's group T also has property R_\infty.