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For a punctured surface $\mathfrak{S}$, the author and Scarinci have recentlyconstructed a quantization of a moduli space of Lorentzian metrics on the3-manifold $\mathfrak{S} \times \mathbb{R}$ of constant sectional curvature$\Lambda \in \{-1,0,1\}$. The invariance of this quantization under the actionof the mapping class group ${\rm MCG}(\mathfrak{S})$ of $\mathfrak{S}$ yieldsfamilies of unitary representations of ${\rm MCG}(\mathfrak{S})$ on a Hilbertspace, with key ingredients being three versions of the quantum dilogarithmfunctions depending on $\Lambda$. In this survey article, we review andelaborate on this result.

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