We develop a systematic method for analyzing the causal structure at vertices in (2+1)-dimensional Lorentzian simplicial gravity. By examining the intersection patterns of lightcones emanating from a vertex with its simplicial neighbourhood, we identify 13 distinct causal types of Lorentzian tetrahedra-excluding configurations with null faces. This forms the basis for a topological classification of the local causal structure in terms of the number of connected spacelike and timelike regions on the triangulated 2-sphere surrounding the vertex. These local causal data allow us to identify regular causal structures and new types of causal irregular configurations, generalizing well-known (1+1)-dimensional topologies, such as `Trousers' and `Yarmulkes', to higher dimensions. We further investigate the dynamical implications of vertex causality by analyzing the behaviour of deficit angles and the Regge action in explicit configurations. Transitions in vertex causal structure coincide with singularities in the curvature and action, suggesting discrete topology change. Causally irregular hinges correspond to discrete conical singularities, while vertex causal irregularities manifest as point-like singularities. Interestingly, we find that vertex and hinge causality are generally independent. These results have direct implications for discrete quantum gravity approaches where the emergence of semiclassical spacetime depends on how causal structures are encoded within the triangulation.
Asante et al. (Mon,) studied this question.