On closed subgroups of the R. Thompson group F

We prove that Thompson’s group F has a subgroup H such that the conjugacy problem in H is undecidable and the membership problem in H is easily decidable. The subgroup H of F is a closed subgroup of F. That is, every function in F which is a piecewise-H function belongs to H. Other interesting examples of closed subgroups of F include Jones’ subgroups F→n and Jones’ 3-colorable subgroup F. By a recent result of the first author, all maximal subgroups of F of infinite index are closed. In this paper we prove that if K ≤ F is finitely generated then the closure of K, i.e., the smallest closed subgroup of F which contains K, is finitely generated. We also prove that all finitely generated closed subgroups of F are undistorted in F. In particular, all finitely generated maximal subgroups of F are undistorted in F.