Branch Groups

Abstract

This chapter discusses the abstract branch groups. There are two new approaches to the definition of a branch group. The first one is purely algebraic, defining branch groups as groups whose lattice of subnormal subgroups is similar to the structure of a spherically homogeneous rooted tree. The second one is based on a geometric point of view according to which branch groups are groups acting spherically transitively on a spherically homogeneous rooted tree and having structure of subnormal subgroups similar to the corresponding structure in the full group Aut (T) of automorphisms of the tree. The main features of a general method which works for almost any finitely generated branch group had appeared: one considers the stabilizer of a vertex on the first level and projects it on the corresponding subtree. The class of branch groups is one of the classes in which the class of just-infinite groups naturally splits. Branch groups have many applications and are related to analysis, geometry, combinatorics, probability, and computer science. The class contains groups with many extraordinary properties, like infinite finitely generated torsion group and groups of intermediate growth.