The dilogarithmic central extension of the Ptolemy–Thompson group via the Kashaev quantization

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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 TˆKash 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⁎, Tab♯ of T by Z, which are of topological nature. We show TˆKash≅Tab♯. Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension TˆCF of T resulting from the Chekhov–Fock(–Goncharov) quantization and thus showed that it corresponds to 12 times the Euler class and that TˆCF≅Tab⁎. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.