Degenerations of ideal hyperbolic triangulations

Let M be a cusped 3-manifold, and let ${mathcal{T}}$ be an ideal triangulation of M. The deformation variety ${mathfrak{D}(mathcal{T})}$ , a subset of which parameterises (incomplete) hyperbolic structures obtained on M using ${mathcal{T}}$ , is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.