On subgroups of R. Thompson's group $F$

TL;DR

This paper demonstrates the existence of maximal subgroups of infinite index in R. Thompson's group F that do not fix any point in the interval, using two different methods, and explores subgroup sequences with specific intersection properties.

Contribution

It introduces two novel methods to construct and analyze maximal subgroups of infinite index in F, solving a problem posed by D. Savchuk.

Findings

Existence of maximal subgroups of infinite index in F that do not fix any point.

Explicit finitely generated examples of such subgroups.

A decreasing sequence of finitely generated subgroups with trivial intersection.

Abstract

We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones' subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings' core graphs, and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>...$ such that $\cap H_i=\{1\}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.