the tetrahedra $T_5$ and $T_9$, in Sec. \[cdt\] referred to as (3,1)- and (2,2)-tetrahedra respectively.

In the present work we will for reasons of simplicity investigate the version of the model where causal spacetimes are assembled from the tetrahedral types $T_2$, $T_3$, $T_5$ and $T_9$ (without $T_7$). As we will see, this already serves our purpose of breaking up the fixed foliated structure.

In 2+1 dimensions violations of local causality – which should therefore be forbidden by the gluing rules – can in principle occur at the links and the vertices of a triangulation. To check whether the light cone structure at a given link is well-behaved, it is sufficient to consider the geometry of a two-dimensional piecewise flat surface orthogonal to the link at its midpoint. This geometry is completely characterized by the set of tetrahedra sharing the link, the so-called star of the link (Fig. \[fig:causality3d\], centre). Each tetrahedron in the star contributes a dihedral angle, defined by the intersection of the tetrahedron with the plane perpendicular to the link (Fig. \[fig:causality3d\], left). The plane segments spanned by all the dihedral angles associated to the given link form a new plane[^7], as shown in Fig. \[fig:causality3d\] (right).

We 