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 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.