Left-Invariant Riemannian Distances on Higher-Rank Sol-Type Groups

In this paper, we generalize the results of ($\mathit{\text{Groups,}}\mathit{\text{Geom.}}\mathit{\text{Dyn.}}$, forthcoming) to describe the split left-invariant Riemannian distances on higher-rank Sol-type groups $G=\mathbf{N}⋊{\mathbb{R}}^k$. We show that the rough isometry type of such a distance is determined by a specific restriction of the metric to ${\mathbb{R}}^k$, and therefore the space of rough similarity types of distances is parameterized by the symmetric space $S{L}_k(\mathbb{R})\mathrm{/}S{O}_k(\mathbb{R})$. In order to prove this result, we describe a family of uniformly roughly geodesic paths, which arise by way of the new technique of $\mathit{\text{Euclidean}}\mathit{\text{curve}}\mathit{\text{surgery}}$.