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## Abstract

Results in C

^{∗}algebras, of Matte Bon and Le Boudec, and of Haagerup and Olesen, apply to the R. Thompson groups F ≤ T ≤ V. These results together show that F is non-amenable if and only if T has a simple reduced C^{∗}-algebra. In further investigations into the structure of C^{∗}-algebras, Breuillard, Kalantar, Kennedy, and Ozawa introduce the notion of a normalish subgroup of a group G. They show that if a group G admits no non-trivial finite normal subgroups and no normalish amenable subgroups then it has a simple reduced C^{∗}-algebra. Our chief result concerns the R. Thompson groups F < T < V; we show that there is an elementary amenable group E < F (where here, E ≅ ...)≀Z)≀Z)≀Z) with E normalish in V. The proof given uses a natural partial action of the group V on a regular language determined by a synchronizing automaton in order to verify a certain stability condition: once again highlighting the existence of interesting intersections of the theory of V with various forms of formal language theory.Original language | English |
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Title of host publication | Developments in Language Theory |

Subtitle of host publication | 24th International Conference, DLT 2020, Tampa, FL, USA, May 11–15, 2020, Proceedings |

Editors | Nataša Jonoska, Dmytro Savchuk |

Publisher | Springer |

Chapter | 3 |

Pages | 29-42 |

Number of pages | 14 |

ISBN (Electronic) | 9783030485160 |

ISBN (Print) | 9783030485153 |

DOIs | |

Publication status | Published - May 2020 |

Event | 24th International Conference on Developments in Language Theory (DLT) - Tampa, United States Duration: 11 May 2020 → 15 May 2020 Conference number: 24 https://www.usf.edu/arts-sciences/conferences/dlt2020/index.aspx |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer |

Volume | 12086 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 24th International Conference on Developments in Language Theory (DLT) |
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Abbreviated title | DLT |

Country/Territory | United States |

City | Tampa |

Period | 11/05/20 → 15/05/20 |

Internet address |

## Keywords

- Thompson's group
- Amenable
- C*-simplicity
- Regular language
- Synchronizing automata
- Group actions
- Normalish sub-groups
- Wreath product

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