This survey concentrates on the use of Lie-theoretic methods in the study of pro-\(p\)-groups. Some related areas are also discussed such as finite \(p\)-groups and residually finite groups. The author focuses on most recent developments and applications.
Here we can only briefly indicate the topics considered in this survey. Chapter 1 is an introduction. Chapter 2 is about Burnside-type problems and the work of Zelmanov, which, on the one hand, solved the classical Restricted Burnside Problem and developed machinery for solving a range of other related questions. The author mentions several such applications, e. g. to \(n\)-Engel groups satisfying certain finiteness conditions (Wilson-Zelmanov, Burns-Macedońska-Medvedev), and to probabilistic identities (Lévai-Pyber). Zel’manov’s theorem can also be combined with recent results of Bakhturin-Zaicev-Linchenko on polynomial identities of Lie algebras with automorphisms to prove results on (pro) \(p\)-groups (Shalev, Shumyatsky), as discussed in Chapter 3. In Chapter 3 also applications of the fixed-point theorems on Lie algebras (Higman, Kreknin, Kostrikin) to (pro) \(p\)-groups with automorphisms are discussed (Khukhro, Medvedev, Shalev). Some of the results required developing new ways of associating the Lie ring with a finite \(p\)-group based either on powerful \(p\)-groups or on the Baker-Campbell-Hausdorff formula.
[Reviewer’s remark: Quite recently A. Jaikin-Zapirain [Adv. Math. 153, No. 2, 391-402 (2000; Zbl 0968.20011)] has solved the reviewer’s problem by proving that in a finite \(p\)-group with an automorphism of order \(p^n\) and exactly \(p^m\) fixed points there exists a subgroup of bounded index, which is soluble of \(m\)-bounded derived length. Earlier, Medvedev solved this in the case \(m=1\) and also provided reduction to Lie rings in the general case.]
Chapter 4 is on Hausdorff dimension and Kac-Moody algebras. The Hausdorff spectrum for profinite groups is defined and some of its properties are mentioned (Abercrombie). Then this spectrum is discussed for various types of pro-\(p\)-groups: \(p\)-adic analytic (Barnea-Shalev, Lévai), some \(\mathbb{F}_p[[t]]\)-analytic ones (Barnea-Shalev), Kac-Moody algebras (Barnea-Shalev-Zelmanov), the Nottingham group (Barnea-Klopsch), free pro-\(p\)-groups, and some others. Chapter 5 is on analytic groups, algebraic groups and subgroup structure. Here the questions on whether certain pro-\(p\)-groups, like the free ones, the Nottingham group, the Grigorchuk groups, are analytic or algebraic are discussed (Pink, Barnea-Larsen, Zubkov). Chapter 6 is on growth of pro-\(p\)-groups, which is here understood either as the growth of the ranks (or orders) of the factors of some filtration, or as the subgroup growth. Some kind of classification relative to given growth is an ultimate goal in such studies. Coclass theory (Leedham-Green, Donkin, McKay, Plesken, and others) is described, with simpler proofs and effectivizations that came from using the Lie theoretic approach (Shalev, Zelmanov). Problems for Lie algebras of given coclass can be more general, since only a small number of them “comes from” groups (Shalev, Zelmanov, Caranti-Mattarei-Newman). Pro-\(p\)-groups of finite width are discussed, which include the Nottingham group and Grigorchuk’s groups. Then Lie algebras of Gel’fand-Kirillov dimension 1 are discussed and the growth-type results for them (Mathieu, Martinez-Zelmanov). Then subgroup growth results are discussed (du Sautoy, Shalev, Segal). The survey also contains numerous conjectures and open problems which outline the current trends in the area and possible lines of attack.
For the entire collection see [Zbl 0945.00009].