In 1980 the author of this chapter constructed and studied some remarkable new examples of finitely generated torsion groups. These groups are residually finite and amenable \(p\)-groups. Five years later Grigorchuk solved a problem of Milnor by showing that their growth is faster than polynomial growth and slower than exponential growth. In 1983 a similar construction of 2-generated infinite \(p\)-groups was obtained by Gupta and Sidki. These two constructions and their variations give a big class of infinite finitely generated torsion groups. This class is called the class of branch groups in the chapter.
The present chapter can be considered as an introduction to the theory of branch groups. The main construction is given and concrete important examples are considered in details. As it is shown branch groups can be viewed via their action on a spherically homogeneous rooted tree of bounded valency. This already implies that they are residually finite. Moreover, the group of automorphisms of such a tree is profinite. This gives the profinite part of the theory. The definitions of the abstract and profinite branch groups are given at the same time in Section 5.
In fact the author concentrates on just infinite branch groups (i.e., groups all whose quotients are finite) in the chapter. It is clear that an infinite group can be mapped onto a just infinite group and as shown in Section 6 using a Frattini argument every pro-\(p\) group can be mapped onto a just infinite pro-\(p\) group. Furthermore, the class of just infinite groups splits into three subclasses: branch groups, virtually simple groups, and hereditarily just infinite groups (i.e., groups all whose subgroups of finite index are just infinite). In the profinite case there are two classes only: simple infinite groups do not appear there. This is shown in Section 6. Section 7 contains a criterion to detect when a branch group (a profinite branch group) is just infinite. The main construction of the branch group in terms of its action on a rooted tree is described in section 8. Section 9 gives a sufficient condition and a criterion to determine when a branch group given by the main construction is torsion.
The connection between abstract and profinite branch groups is discussed in Section 10. Namely, the congruence property for a group acting on a rooted tree \(T\) is introduced; this establishes a connection between the profinite completion of a branch group and their closure in \(\operatorname{Aut}(T)\).
In Section 11 the author shows that the profinite completion of a branch \(p\)-group satisfying the congruence property is very large: it contains every second countable (in particular every finitely generated) pro-\(p\) group.
The final four sections deal with the first example of a torsion branch group \(G\) constructed by the author, a 3-generated residually finite 2-group. Section 12 contains general information, in Section 13 the author describes the derived series and in Section 14 the central series of this group. In particular, it is shown that the quotients \(\gamma_n(G)/\gamma_{n+1}(G)\) are elementary Abelian groups of rank bounded by 2 when \(n>1\).
Finally it is shown in Section 15 that the Hausdorff dimension of \(\widehat G\) as a subgroup of the automorphism group of a binary tree is \(5/8\).
For the entire collection see [Zbl 0945.00009].