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

Abstract

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.