The book is dedicated to the following question: given a group acting on a median graph, or equivalently on a CAT(0) cube complex, how are related the algebraic properties of the groups with the geometric properties of the graph? 

Not all the chapters are yet written. This is a draft. More complete versions will follow.

Version 1, 1st Septembre, 2021.                                                                                         Version 2, 6th August, 2022. Chapters 8, 9, 10, 11, 12 and Section 2.4 added.                                                                                                                                                    Version 3, 18th June 2023. Theorem 6.3.1 improved, proof of Theorem 5.1.3 corrected, Section 11.1 rewritten as Sections 11.1 and 11.2, Section 0.5.1 added, Chapter 16 added.                      Version 4, 9th Septembre, 2024. Chapters 15, 18, 20, 21, 22, and 23 added.