R. I. Grigorchuk has introduced the class of branch groups [in: New horizons in pro-\(p\) groups, Prog. Math. 184, 121-179 (2000; Zbl 0982.20024)]. Branch groups are defined as groups of automorphisms of rooted, spherically homogeneous trees. They include many examples of torsion groups, groups of intermediate growth and groups of finite width with striking properties. Branch groups extend constructions of R. I. Grigorchuk [see for instance Funkts. Anal. Prilozh. 14, No. 1, 53-54 (1980; Zbl 0595.20029)] and N. Gupta and S. Sidki [Math. Z. 182, 385-388 (1983; Zbl 0513.20024)].
In the paper under review the authors study the profinite completions of branch groups. They prove the remarkable fact that every countably based pro-\(p\)-group can be embedded in the completion of a branch group, provided there is no prime \(p\) for which the latter is virtually \(p\)-torsion free. This can be compared with other embedding theorems of A. Lubotzky and J. S. Wilson [Arch. Math. 42, 397-399 (1984; Zbl 0522.20023)], R. Camina [J. Algebra 196, No. 1, 101-113 (1997; Zbl 0883.20015)] and Z. Chatzidakis [J. Group Theory 2, No. 1, 65-68 (1999; Zbl 0923.20019)].
For the entire collection see [Zbl 0933.00023].