Geometric and causal properties of the sub-Lorentzian Heisenberg group

In this talk we present the sub-Lorentzian geometry of the Heisenberg group $\mathbb{H}$, the simplest non-holonomic analogue of a spacetime. After outlining motivations that show why such structure is a compelling object of study, we analyze the causal structure and present a characterization of geodesics of the space via a planar Lorentzian isoperimetric problem. We further explain geometrically why this space does not admit standard synthetic Ricci curvature bounds. If time permits, we will also discuss its Lorentzian Hausdorff dimension which we compute to be $4$, analogously to the usual Hausdorff dimension of the space and therefore in contrast with the topological dimension of the manifold which is $3$. This talk is based on joint work with Chiara Rigoni and Samuël Borza.