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

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 ZZ, 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)H2(T;Z). Meanwhile, the braided Ptolemy–Thompson groups T⁎T⁎, T♯T♯ of Funar–Kapoudjian are extensions of T   by the infinite braid group B∞B∞, and by abelianizing the kernel B∞B∞ one constructs central extensions Tab⁎, Tab♯ of T   by ZZ, 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.