Causal structure and topology change in ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> )-dimensional simplicial gravity

We develop a systematic method for analyzing the causal structure at vertices in ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mn>2</a:mn> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:mrow> </a:math> )-dimensional Lorentzian simplicial gravity. By examining the intersection patterns of light ones emanating from a vertex with its simplicial neighborhood, we identify 13 distinct causal types of Lorentzian tetrahedra—excluding configurations with null faces. This classification forms the basis for a topological characterization of the local causal structure in terms of the number of connected regions on the triangulated 2-sphere that are spacelike and timelike separated from a bulk vertex. These local causal data allow us to identify regular causal configurations and new types of irregular causal configurations, generalizing well-known ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:mn>1</c:mn> <c:mo>+</c:mo> <c:mn>1</c:mn> </c:mrow> </c:math> )-dimensional topologies, such as “Trousers” and “Yarmulkes,” to higher dimensions. We further investigate the dynamical implications of vertex causality by analyzing the behavior of deficit angles and the Regge action in explicit configurations. Transitions in “vertex causality” coincide with discontinuities in the curvature, suggesting discrete topology change. Causally irregular hinges correspond to discrete conical singularities, while vertex causal irregularities manifest as pointlike 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.