On closed subgroups of the R. Thompson group F

Gili Golan, Mark Sapir

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Abstract

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 KF 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 languageEnglish
JournalArxiv preprint
StatePublished - 2021

Keywords

  • math.GR

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