A primer on Carnot groups: homogenous groups, CC spaces, and regularity of their isometries

TL;DR

This paper provides an overview of Carnot groups, their structure as homogeneous metric spaces, and investigates the regularity properties of their isometries, contributing to the understanding of their geometric and algebraic features.

Contribution

It offers a comprehensive introduction to Carnot groups, including their classification as graded and homogeneous spaces, and analyzes the regularity of isometries in these contexts.

Findings

Carnot groups are Lie groups with invariant path distances and dilation automorphisms.

The paper discusses the regularity of isometries in Carnot-Carathéodory spaces.

It clarifies the structure of nilpotent metric Lie groups and their isometries.

Abstract

Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Caratheodory spaces and of nilpotent metric Lie groups.