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

TL;DR

This paper characterizes the space of left-invariant Riemannian distances on higher-rank Sol-type groups, showing they are classified by a symmetric space and introducing a new Euclidean curve surgery technique.

Contribution

It generalizes previous results to higher-rank groups and introduces Euclidean curve surgery to analyze geodesic paths and distance classifications.

Findings

Rough isometry types are determined by metrics on a0^k

The space of distances is parameterized by SL_k( Real)/SO_k( Real)

Introduces Euclidean curve surgery for path analysis

Abstract

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 $\textit{Euclidean curve surgery}$.