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

Quantization of universal Teichm\"uller 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 $\mathbb{Z}$, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension $\hat{T}^{Kash}$ of $T$ resulting from the Kashaev quantization, and show that it corresponds to $6$ times the Euler class in $H^2(T;\mathbb{Z})$. Meanwhile, the braided Ptolemy-Thompson groups $T^*$, $T^\sharp$ of Funar-Kapoudjian are extensions of $T$ by the infinite braid group $B_\infty$, and by abelianizing the kernel $B_\infty$ one constructs central extensions $T^*_{ab}$, $T^\sharp_{ab}$ of $T$ by $\mathbb{Z}$, which are of topological nature. We show $\hat{T}^{Kash}\cong T^\sharp_{ab}$. Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension $\hat{T}^{CF}$ of $T$ resulting from the Chekhov-Fock(-Goncharov) quantization and thus showed that it corresponds to $12$ times the Euler class and that $\hat{T}^{CF} \cong T^*_{ab}$. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.