We study the spectral geometric properties of the scalar Laplace–Beltrami operator
associated to the Weil–Petersson metric $g_{WP}$ on $\mathcal{M}_{\gamma}$ , the Riemann moduli space of surfaces of
genus $\gamma>1$. This space has a singular compactification with respect to $g_{WP}$, and this metric
has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors.
We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its
spectrum is discrete.