The CD condition conjecture for sub-Finsler Carnot groups

A sub-Finsler Carnot group is a stratified, nilpotent Lie group equipped with a left-invariant distance. Its homogeneous dimension is denoted by $\mathcal{Q}$, and $\mathcal H^\mathcal{Q}$ denotes the Hausdorff measure in that dimension. The CD condition conjecture. Let $(G,\mathsf d_{SF})$ be a sub-Finsler Carnot group of homogeneous dimension $\mathcal{Q}$. Assume that the metric measure space $(G,\mathsf d_{SF},\mathcal H^\mathcal{Q})$ is a $\mathsf{CD}(0,N)$ space for some $N>1$. Then $G$ is a finite-dimensional Banach space, i.e. $G$ is commutative, $\mathfrak g=V_1$ and $\mathcal{Q}=\dim_{\rm Top}G$. The conjecture would extend known low-dimensional results to arbitrary homogeneous dimension and would imply the paper’s general rectifiability theorem for spaces with unique tangents. It is supported by results in the sub-Riemannian and sub-Finsler settings, but remains open in general.