Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler ${\mathrm{\ell }}^p$-Heisenberg groups

We initiate the study of synthetic curvature-dimension bounds in sub-Finsler geometry. More specifically, we investigate the measure contraction property $\mathsf{M}\mathsf{C}\mathsf{P}(K,N)$, and the geodesic dimension on the Heisenberg group equipped with an ${\mathrm{\ell }}^p$-sub-Finsler norm. We show that for $p\in (2,\mathrm{\infty }]$, the ${\mathrm{\ell }}^p$-Heisenberg group fails to satisfy any of the measure contraction properties. On the other hand, if $p\in (1,2)$, then it satisfies the measure contraction property $\mathsf{M}\mathsf{C}\mathsf{P}(K,N)$ if and only if $K\le 0$ and $N\ge N_p$, where the curvature exponent $N_p$ is strictly greater than $2q+1$ ($q$ being the Hölder conjugate of $p$). We also prove that the geodesic dimension of the ${\mathrm{\ell }}^p$-Heisenberg group is $\min (2q+2,5)$ for $p\in [1,\mathrm{\infty })$. As a consequence, we provide the first example of a metric measure space where there is a gap between the curvature exponent and the geodesic dimension.