MCP and geodesic dimension on the $l^p$ Heisenberg group

We study the measure contraction property $MCP(0,N)$ and the geodesic dimension on the Heisenberg group with the $l^p$ sub-Finsler metric. We show that if $p$ is in $(2,\infty]$, then it fails to be $MCP(0,N)$. On the other hand, if $p$ is in $(1,2)$, then it satisfies $MCP(0,N)$ with $N$ strictly greater than $2q+1$ (q being the Hölder conjugate). Furthermore, the geodesic dimension is explicitly given by $\min\{2q+2,5\}$ for all $p$ in $[1,\infty)$. When $p$ is in $(1,\infty)$, our technique is based on the Taylor expansion of the generalized trigonometric function. If $p$ is $1$ or infinity, then its branching geodesics and cut locus are explicitly computed and it yields the conclusion.
This is a joint work with Samuel Borza (SISSA). We put the preprint on the following arXiv link.
https://arxiv.org/abs/2305.16722