Metric Lie Groups

This Open Access textbook presents Carnot-Carathéodory spaces from the perspective of Lie groups. Its main objective is to illustrate how these non-smooth geometries manifest in various mathematical domains, including metric geometry and geometric group theory. In contrast to other sources, this book utilizes the formalism of Lie groups to showcase how this theory facilitates the development of geometry and analysis on the non-smooth structure of Carnot-Carathéodory spaces. Major results are presented with rigorous mathematical proofs, and references for further exploration are provided. Open problems in these areas are discussed, offering insights into recent developments and avenues for future research. Prerequisite topics such as differential geometry, measure theory, and group theory are incorporated in the main flow of the chapters, ensuring a comprehensive understanding. Junior researchers seeking an introduction to the field of sub-Riemannian geometry will find this an invaluable introductory companion. The book is also suitable for those entering research subjects on the interplay between geometry, analysis, and group theory.