Left-invariant Riemannian distances on higher-rank Sol-type groups

Abstract

In this paper, we generalize the results of Le Donne, Pallier, and Xie [Groups Geom. Dyn. 19 (2025), 227–263] 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 ${\mathrm{SL}}_k(\mathbb{R})/{\mathrm{SO}}_k(\mathbb{R})$. In order to prove this result, we describe a family of uniformly roughly geodesic paths, which arise by way of the new technique of Euclidean curve surgery.