Hausdorff dimension and failure of synthetic curvature bounds in the sub-Lorentzian Heisenberg group

We study the geodesics, Hausdorff dimension, and curvature bounds of the sub-Lorentzian Heisenberg group. Through an elementary variational approach, we provide a new proof of the structure of its maximizing geodesics, showing that they are lifts of hyperbolae coming from a Lorentzian isoperimetric problem in the Minkowski plane. We prove that the Lorentzian Hausdorff dimension of the space is $4$ and that the corresponding measure coincides with the Haar measure. We further establish a novel result in the spirit of the Ball-Box theorem, giving a uniform estimate of causal diamonds by anisotropic boxes. Finally, we show that the Heisenberg group satisfies neither the timelike curvature-dimension condition $\mathsf{TCD}(K,N)$ nor the timelike measure contraction property $\mathsf{TMCP}(K,N)$ for any values of the parameters $K$ and $N$, in sharp contrast with its sub-Riemannian counterpart.