On closed subgroups of the R. Thompson group $F$

TL;DR

This paper explores the structure of closed subgroups within Thompson's group F, revealing properties about their generation, distortion, and the complexity of problems like conjugacy and membership within these subgroups.

Contribution

It demonstrates that finitely generated subgroups have finitely generated closures and are undistorted, and constructs a subgroup with an undecidable conjugacy problem but decidable membership.

Findings

Existence of a subgroup with undecidable conjugacy problem and decidable membership

Finitely generated subgroups have finitely generated closures

All finitely generated closed subgroups are undistorted in F

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 $\overrightarrow{F}_n$ and Jones' $3$-colorable subgroup $\mathcal F$. 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 $K\leq F$ 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 $F$.