## Abstract

Quantization of universal Teichmüller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group T. This yields certain central extensions of T by Z, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension T Kash of T resulting from the Kashaev quantization, and show that it corresponds to 6 times the Euler class in H2(T;Z). Meanwhile, the braided Ptolemy-Thompson groups T*, T of Funar-Kapoudjian are extensions of T by the infinite braid group B∞, and by abelianizing the kernel B∞ one constructs central extensions Tab*, T_{ab} of T by Z, which are of topological nature. We show T Kash≅Tab . Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension T CF of T resulting from the Chekhov-Fock(-Goncharov) quantization and thus showed that it corresponds to 12 times the Euler class and that T ^{CF}≅Tab*. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.

Original language | English |
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Pages (from-to) | 529-588 |

Number of pages | 60 |

Journal | Advances in Mathematics |

Volume | 293 |

DOIs | |

State | Published - 30 Apr 2016 |

## Keywords

- Braided Ptolemy-Thompson group
- Infinite braid group
- Kashaev quantization
- Ptolemy-Thompson group
- Quantum Teichmüller theory
- Stable braid group
- Thompson group T
- Universal Teichmüller space