In this paper, we generalize the results of ($\textit{ Groups, Geom. Dyn.}$, forthcoming) to describe the split left-invariant Riemannian distances on higher-rank Sol-type groups $G=\mathbf{ N}\rtimes \mathbb{ R}^k$. 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 $SL_k(\mathbb{ R})/SO_k(\mathbb{ R})$. In order to prove this result, we describe a family of uniformly roughly geodesic paths, which arise by way of the new technique of $\textit{ Euclidean curve surgery}$.
Categories: math.GR math.DG math.MG