Burnside Groups and Groups Acting on Rooted Trees

Branch groups stem from one of the most influential problems in group theory: The famous Burnside Problem, which arose in 1902 and asks if a finitely generated torsion group, i.e. in which every element has finite order, can be itself infinite. Now branch groups are groups acting on rooted trees, that have a rich tree-like subgroup structure. There are many examples of branch groups with remarkable algebraic properties, and branch groups have many applications within group theory and also to other areas of mathematics, such as to dynamics, analysis, algebraic geometry and cryptography. These lecture notes aim to introduce branch groups and some of their well-studied generalisations. An overview of the wide array of applications of branch groups will be given, including their use to answer several open questions. We will then focus on some new developments and some big open problems in the subject, such as those concerning maximal subgroups of branch groups.