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Golod-Shafarevich groups: a survey. (English) Zbl 1286.20033
The class field tower problem was posed by Furtwängler in 1925, and popularised by H. Hasse [Jahresbericht D. M. V. 35, 1-55 (1926; JFM 52.0150.19)]. It can be stated as saying that there is no number field \(K\) whose maximal unramified prosoluble extension has infinite degree over \(K\). To provide a negative answer, one could show that for some prime \(p\) the maximal unramified \(p\)-extension \(K_p\) of \(K\) has infinite Galois group \(G_{K,p}\), the latter being a pro-\(p\) group.
I. R. Shafarevich gave in 1963 [Publ. Math., Inst. Hautes Étud. Sci. 18, 295-319 (1963; Zbl 0118.27505)] a formula for the minimal number of generators \(d(G_{K,p})\) of \(G_{K,p}\), and an upper bound for the minimal number \(r(G_{K,p})\) of relations. These results implied a negative answer to the problem, if one could show that there is no infinite sequence of finite \(p\)-groups \(G_n\) such that \(d(G_n)\to\infty\), while \(r(G_n)-d(G_n)\) stays bounded. This was achieved by E. S. Golod and I. R. Shafarevich in 1964 [Izv. Akad. Nauk SSSR Ser. Mat. 28, 261-272 (1964; Zbl 0136.02602)], when they were able to show that if \(G\) is a finite \(p\)-group, then \(r(G)>(d(G)-1)^2/4\), a result later improved to \(r(G)>d(G)^2/4\) by È. B. Vinberg [Izv. Akad. Nauk SSSR, Ser. Mat. 29, 209-214 (1965; Zbl 0171.29401)] and P. Roquette [in Algebraic Number Theory. Proceedings of an instructional conference organized by the London Mathematical Society. 231-249 (1967; Zbl 0153.07403)].
The paper under review is a survey of Golod-Shafarevich groups and algebras, that is, those groups and algebras \(G\) that satisfy the Golod-Shafarevich inequality \(r(G)>d(G)^2/4\). Here the relators can be counted in a certain weighted sense. E. S. Golod was able to show [Am. Math. Soc., Translat., II. Ser. 48, 103-106 (1965); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 28, 273-276 (1964; Zbl 0215.39202)] that there are Golod-Shafarevich abstract groups that are torsion, thus settling in the negative also the general Burnside problem.
Section 2 deals with the Golod-Shafarevich inequality for graded algebras, which is formulated in terms of Hilbert series and its applications to settling in the negative the Kurosh-Levitzky problem and the general Burnside problems.
Section 3 gives a formal definition of Golod-Shafarevich groups via degree functions and Hilbert series. A stronger version of a result of Golod is proved, stating that for every prime \(p\) and integer \(d\geq 2\) there exists an infinite \(d\)-generated \(p\)-torsion group in which every \((d-1)\)-generated subgroup is finite.
Sections 4 and 5 deal with generalised Golod-Shafarevich groups, which were introduced by M. Ershov and A. Jaikin-Zapirain [in J. Reine Angew. Math. 677, 71-134 (2013; Zbl 1285.20031)]. Here the generators, too, are counted with weights, and their number is allowed to be countable.
Quotients of generalised Golod-Shafarevich groups provide a wide range of examples of infinite groups with specific properties. The quotients, and several of their applications, are dealt with in Section 6. For instance, a result of A. Myasnikov and D. Osin [J. Pure Appl. Algebra 215, No. 11, 2789-2796 (2011; Zbl 1233.20031)] can be derived, which shows that every recursively presented Golod-Shafarevich abstract group has a Golod-Shafarevich quotient \(Q\) such that there is no algorithm that can produce an infinite set of pairwise distinct elements of \(Q\).
In Section 7 an important result of E. Zelmanov [in: New horizons in pro-\(p\) groups. Prog. Math. 184, 223-232 (2000; Zbl 0974.20022)] is discussed, which shows that every generalised Golod-Shafarevich pro-\(p\) group contains a nonabelian free pro-\(p\) subgroup.
Section 8 deals with subgroup growth.
Section 9 discusses the recent discovery of very simple counterexamples to the general Burnside problem by J.-C. Schlage-Puchta [J. Group Theory 15, No. 2, 261-270 (2012; Zbl 1259.20046)] and D. Osin [Bull. Lond. Math. Soc. 43, No. 1, 10-16 (2011; Zbl 1245.20044)]. These are based on groups of positive power \(p\)-deficiency.
Section 10 surveys applications to number theory, covering in particular the class field tower problem.
Section 11 deals with applications in geometry and topology, in particular concerning the fundamental groups of hyperbolic \(3\)-manifolds.
Section 12 discusses the work by the author on Kazhdan’s property (T) for Golod-Shafarevich groups [Duke Math. J. 145, No. 2, 309-339 (2008; Zbl 1162.20018); Proc. Lond. Math. Soc. (3) 102, No. 4, 599-636 (2011; Zbl 1280.20037)].
Section 13 discusses a result of the author and A. Jaikin-Zapirain [loc. cit.], which states that for every prime \(p\), and every abstract generalised Golod-Shafarevich \(G\) with respect to \(p\), there is a quotient of \(G\) which is a finitely generated, residually finite, infinite \(p\)-torsion group in which every finitely generated subgroup is either finite, or of finite index. This is related to the Tarski Monsters, first constructed by A. Yu. Ol’shanskiĭ [Math. USSR, Izv. 16, 279-289 (1981); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 309-321 (1980; Zbl 0475.20025)].
Section 14 closes with a discussion of several open problems.
The author is a leading expert in the field, to which he has given fundamental contributions. This well-written, comprehensive survey is to be recommended to everyone interested in the area.

MSC:
20F05 Generators, relations, and presentations of groups
20F50 Periodic groups; locally finite groups
20E18 Limits, profinite groups
20E07 Subgroup theorems; subgroup growth
20F69 Asymptotic properties of groups
16W50 Graded rings and modules (associative rings and algebras)
17B50 Modular Lie (super)algebras
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