Coarse differentiation and quasi-isometries of a class of solvable Lie groups I

TL;DR

This paper investigates the quasi-isometric properties of a specific subclass of unimodular split solvable Lie groups, demonstrating that such maps are approximately structure-preserving at a local scale.

Contribution

It establishes that locally, quasi-isometries between these Lie groups are close to maps that respect their algebraic structures, advancing understanding of their geometric rigidity.

Findings

Quasi-isometries are close to structure-preserving maps locally.

Provides foundational results for classifying solvable Lie groups via coarse geometry.

First in a series exploring quasi-isometric rigidity of these groups.

Abstract

This is the first of two papers which aim to understand quasi-isometries of a subclass of unimodular split solvable Lie groups. In the present paper, we show that locally (in a coarse sense), a quasi-isometry between two groups in this subclass is close to a map that respects their group structures.