## Abstract

We prove that Thompson’s group

We also prove that all finitely generated closed subgroups of

*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.Original language | English |
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State | Published - 2021 |

## Keywords

- math.GR

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