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

Abstract

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 $\mathbb{Z}$, 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 $T^{\ast }$, $T^{♯}$ 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 $T_{ab}^{\ast }$, $T_{ab}^{♯}$ of $T$ by $\mathbb{Z}$, which are of topological nature. We show ${\hat{T}}^{Kash}\cong T_{ab}^{♯}$. 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 ${\hat{T}}^{CF}\cong T_{ab}^{\ast }$. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.

43 pages, 15 figures. v2: substantially revised from the first version, and the author affiliation changed. // v3: Groups M and T are shown to be anti-isomorphic (new Prop.2.32), which makes the whole construction more natural. And some minor changes // v4: reflects all changes made for journal publication (to appear in Adv. Math.)

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Cited by in corpus (5)

• Phase constants in the Fock-Goncharov quantum cluster varieties
• Three-dimensional quantum gravity from the quantum pseudo-Kähler plane
• Phase constants in the Fock-Goncharov quantization of cluster varieties: long version
• Irreducible self-adjoint representations of quantum Teichmüller space and the phase constants
• Finite dimensional quantum Teichmüller space from the quantum torus at root of unity