## Abstract

The subgroups H_{U} of the R. Thompson group F that are stabilizers of finite sets U of numbers in the interval (0, 1) are studied. The algebraic structure of H_{U} is described and it is proved that the stabilizer H_{U} is finitely generated if and only if U consists of rational numbers. It is also shown that such subgroups are isomorphic surprisingly often. In particular, if finite sets U ⊂ [0, 1] and V ⊂ [0, 1] consist of rational numbers that are not finite binary fractions, and |U| = |V|, then the stabilizers of U and V are isomorphic. In fact these subgroups are conjugate inside a subgroup F < Homeo([0, 1]) that is the completion of F with respect to what is called the Hamming metric on F. Moreover the conjugator can be found in a certain subgroup F < F which consists of possibly infinite tree-diagrams with finitely many infinite branches. It is also shown that the group F is non-amenable.

Original language | English |
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Pages (from-to) | 51-79 |

Number of pages | 29 |

Journal | St. Petersburg Mathematical Journal |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2018 |

Externally published | Yes |

## Keywords

- Stabilizers
- Thompson group F

## All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Applied Mathematics