## 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 ∩H_{i} = {1} and for every i there exist only finitely many subgroups of F containing H_{i}.

Original language | English |
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Pages (from-to) | 8857-8878 |

Number of pages | 22 |

Journal | Transactions of the American Mathematical Society |

Volume | 369 |

Issue number | 12 |

DOIs | |

State | Published - 1 Dec 2017 |

Externally published | Yes |