A trilogy of mapping class group representations from three-dimensional quantum gravity

For a punctured surface $\mathfrak{S}$, the author and Scarinci (arXiv:2112.13329) have recently constructed a quantization of a moduli space of Lorentzian metrics on the 3-manifold $\mathfrak{S}\times \mathbb{R}$ of constant sectional curvature $\mathrm{\Lambda }\in \{-1,0,1\}$. The invariance of this quantization under the action of the mapping class group $\mathrm{M}\mathrm{C}\mathrm{G}(\mathfrak{S})$ of $\mathfrak{S}$ yields families of unitary representations of $\mathrm{M}\mathrm{C}\mathrm{G}(\mathfrak{S})$ on a Hilbert space, with key ingredients being three versions of the quantum dilogarithm functions depending on $\mathrm{\Lambda }$. In this survey article, we review and elaborate on this result.