Golod-Shafarevich groups: a survey. (Q2909484)

The class field tower problem was posed by Furtwängler in 1925, and popularised by \textit{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.NEWLINENEWLINE \textit{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 \textit{E. S. Golod} and \textit{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 \textit{È. B. Vinberg} [Izv. Akad. Nauk SSSR, Ser. Mat. 29, 209-214 (1965; Zbl 0171.29401)] and \textit{P. Roquette} [in Algebraic Number Theory. Proceedings of an instructional conference organized by the London Mathematical Society. 231-249 (1967; Zbl 0153.07403)].NEWLINENEWLINE 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. \textit{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.NEWLINENEWLINE 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.NEWLINENEWLINE 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.NEWLINENEWLINE Sections~4 and~5 deal with generalised Golod-Shafarevich groups, which were introduced by \textit{M. Ershov} and \textit{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.NEWLINENEWLINE 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 \textit{A. Myasnikov} and \textit{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 \(Q\).NEWLINENEWLINE In Section~7 an important result of \textit{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.NEWLINENEWLINE Section~8 deals with subgroup growth.NEWLINENEWLINE Section~9 discusses the recent discovery of very simple counterexamples to the general Burnside problem by \textit{J.-C. Schlage-Puchta} [J. Group Theory 15, No. 2, 261-270 (2012; Zbl 1259.20046)] and \textit{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.NEWLINENEWLINE Section~10 surveys applications to number theory, covering in particular the class field tower problem.NEWLINENEWLINE Section~11 deals with applications in geometry and topology, in particular concerning the fundamental groups of hyperbolic \(3\)-manifolds.NEWLINENEWLINE 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)].NEWLINENEWLINE Section~13 discusses a result of the author and \textit{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 \textit{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)].NEWLINENEWLINE Section~14 closes with a discussion of several open problems.NEWLINENEWLINE 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.