Metric Lie Groups. Carnot-Carath\'eodory spaces from the homogeneous viewpoint

TL;DR

This book investigates the structure and properties of metric Lie groups, especially Carnot groups, focusing on their geometric and algebraic aspects within sub-Riemannian and sub-Finsler geometries.

Contribution

It provides a comprehensive homogeneous viewpoint on Carnot-Carathéodory spaces, connecting them to various areas like metric geometry and geometric group theory.

Findings

Characterization of geodesic left-invariant metrics as sub-Riemannian or sub-Finsler.

Identification of Carnot groups as asymptotic cones of nilpotent groups.

Description of metric Lie groups as limits and tangents of Riemannian and sub-Riemannian manifolds.

Abstract

This book explores geometries defined by left-invariant distance functions on Lie groups, with a particular focus on nilpotent groups and Carnot groups equipped with geodesic distances. Geodesic left-invariant metrics are either sub-Riemannian or their generalizations, known as sub-Finsler geometries or Carnot-Carath\'eodory metrics. The primary objective is to illustrate how these non-smooth geometries, together with a Lie group structure, manifest in various mathematical fields, including metric geometry and geometric group theory. Additionally, the book demonstrates the role of metric Lie groups, particularly Carnot groups, in the following contexts: (a) as asymptotic cones of nilpotent groups; (b) as parabolic boundaries of rank-one symmetric spaces and, more broadly, of homogeneous negatively curved Riemannian manifolds; (c) as limits of Riemannian manifolds and tangents of sub-Riemannian manifolds.