  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 $\Lambda \in \{-1,0,1\}$. The invariance of this
quantization under the action of the mapping class group ${\rm
MCG}(\mathfrak{S})$ of $\mathfrak{S}$ yields families of unitary
representations of ${\rm MCG}(\mathfrak{S})$ on a Hilbert space, with key
ingredients being three versions of the quantum dilogarithm functions depending
on $\Lambda$. In this survey article, we review and elaborate on this result.
