# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Abert M., Groups Geom. Dyn. 5 pp 213– [2] DOI: 10.1016/0021-8693(82)90104-1 · Zbl 0502.16002 · doi:10.1016/0021-8693(82)90104-1 [3] D. J. Anick, Algebraic Topology and Algebraic K-Theory, Annals of Mathematics Studies 113 (Princeton University Press, Princeton, NJ, 1987) pp. 247–321. [4] DOI: 10.1006/jabr.1998.7682 · Zbl 0923.20018 · doi:10.1006/jabr.1998.7682 [5] DOI: 10.1017/CBO9780511600609.002 · doi:10.1017/CBO9780511600609.002 [6] Baumslag B., J. London Math. Soc. (2) 17 pp 425– [7] DOI: 10.1017/CBO9780511542749 · doi:10.1017/CBO9780511542749 [8] Button J., Groups Geom. Dyn. 4 pp 709– [9] DOI: 10.1142/S0218196711006339 · Zbl 1247.20038 · doi:10.1142/S0218196711006339 [10] DOI: 10.1017/CBO9780511470882 · doi:10.1017/CBO9780511470882 [11] DOI: 10.1007/s00222-002-0242-y · Zbl 1140.20308 · doi:10.1007/s00222-002-0242-y [12] DOI: 10.1093/qmath/12.1.205 · Zbl 0231.55003 · doi:10.1093/qmath/12.1.205 [13] DOI: 10.1215/00127094-2008-053 · Zbl 1162.20018 · doi:10.1215/00127094-2008-053 [14] DOI: 10.1112/plms/pdq022 · Zbl 1280.20037 · doi:10.1112/plms/pdq022 [15] DOI: 10.1007/s00222-009-0218-2 · Zbl 1205.22003 · doi:10.1007/s00222-009-0218-2 [16] DOI: 10.1142/9789812385215_0045 · doi:10.1142/9789812385215_0045 [17] Golod E., Izv. Akad. Nauk SSSR Ser. Mat. 28 pp 273– [18] Golod E., Izv. Akad. Nauk SSSR Ser. Mat. 28 pp 261– [19] Grigorchuk R. I., Funktsional. Anal. i Prilozhen. 14 pp 53– [20] DOI: 10.1006/jabr.1996.6849 · Zbl 0879.11069 · doi:10.1006/jabr.1996.6849 [21] DOI: 10.1017/S030821051099999X · Zbl 1239.16030 · doi:10.1017/S030821051099999X [22] M. I. Kargapolov and Yu. I. Merzlyakov, Osnovy Teorii Grupp [Fundamentals of Group Theory], 4th edn. (Fizmatlit ”Nauka”, Moscow, 1996) p. 288. · Zbl 0884.20001 [23] DOI: 10.1090/S0002-9939-02-06824-7 · Zbl 1024.20026 · doi:10.1090/S0002-9939-02-06824-7 [24] Koch H., Galois Cohomology of Algebraic Number Fields (1978) [25] DOI: 10.1007/978-3-662-04967-9 · doi:10.1007/978-3-662-04967-9 [26] Kostrikin A. I., Izv. Akad. Nauk SSSR Ser. Mat. 29 pp 1119– [27] DOI: 10.1215/S0012-7094-07-13616-0 · Zbl 1109.57015 · doi:10.1215/S0012-7094-07-13616-0 [28] DOI: 10.1112/plms/pdn032 · Zbl 1175.20025 · doi:10.1112/plms/pdn032 [29] DOI: 10.1017/S0305004108002089 · Zbl 1185.57014 · doi:10.1017/S0305004108002089 [30] DOI: 10.1016/j.jalgebra.2010.07.012 · Zbl 1231.20026 · doi:10.1016/j.jalgebra.2010.07.012 [31] Lazard M., Inst. Hautes Études Sci. Publ. Math. 26 pp 389– [32] DOI: 10.1016/j.top.2006.04.001 · Zbl 1169.57003 · doi:10.1016/j.top.2006.04.001 [33] DOI: 10.2307/2006956 · Zbl 0541.20020 · doi:10.2307/2006956 [34] DOI: 10.1112/blms/19.4.325 · Zbl 0597.20043 · doi:10.1112/blms/19.4.325 [35] DOI: 10.1016/0021-8693(82)90006-0 · Zbl 0473.20031 · doi:10.1016/0021-8693(82)90006-0 [36] DOI: 10.1016/0021-8693(83)90005-4 · Zbl 0503.20008 · doi:10.1016/0021-8693(83)90005-4 [37] DOI: 10.2307/2374368 · Zbl 0565.20023 · doi:10.2307/2374368 [38] DOI: 10.1016/0021-8693(87)90212-2 · Zbl 0626.20022 · doi:10.1016/0021-8693(87)90212-2 [39] DOI: 10.1007/BF01472217 · Zbl 0011.15201 · doi:10.1007/BF01472217 [40] Morishita M., J. Reine Angew. Math. 550 pp 141– [41] DOI: 10.1016/j.jpaa.2011.03.019 · Zbl 1233.20031 · doi:10.1016/j.jpaa.2011.03.019 [42] DOI: 10.1007/978-3-540-37889-1 · doi:10.1007/978-3-540-37889-1 [43] DOI: 10.1017/S1446788700013859 · Zbl 0267.20026 · doi:10.1017/S1446788700013859 [44] Novikov P. S., Izv. Akad. Nauk SSSR Ser. Mat. 32 pp 212– [45] Novikov P. S., Izv. Akad. Nauk SSSR Ser. Mat. 32 pp 251– [46] Novikov P. S., Izv. Akad. Nauk SSSR Ser. Mat. 32 pp 709– [47] Ol’shanskii A. Yu., Uspekhi Mat. Nauk 35 pp 199– [48] DOI: 10.1007/978-94-011-3618-1 · doi:10.1007/978-94-011-3618-1 [49] DOI: 10.4007/annals.2010.172.1 · Zbl 1203.20031 · doi:10.4007/annals.2010.172.1 [50] DOI: 10.1112/blms/bdq075 · Zbl 1245.20044 · doi:10.1112/blms/bdq075 [51] DOI: 10.1090/ulect/037 · doi:10.1090/ulect/037 [52] S. Popa and S. Vaes, Quanta of Maths, Clay Mathematics Proceedings 11 (American Mathematical Society, Providence, RI, 2010) pp. 519–541. · Zbl 1222.37008 [53] DOI: 10.1007/s000290050015 · Zbl 0892.57012 · doi:10.1007/s000290050015 [54] P. Roquette, Algebraic Number Theory (Thompson Book, 1967) pp. 231–249. [55] Schmidt A., J. Reine Angew. Math. 596 pp 115– [56] Schmidt A., Doc. Math. 12 pp 441– [57] Schmidt A., J. Reine Angew. Math. 640 pp 203– [58] Schlage-Puchta J.-C., J. Group Theory 15 pp 261– · Zbl 1081.20082 [59] DOI: 10.1007/BF02684785 · doi:10.1007/BF02684785 [60] Vershik A., Selecta Math. Soviet. 2 pp 311– [61] Vinberg E. B., Izv. Akad. Nauk SSSR Ser. Mat. 29 pp 209– [62] DOI: 10.1142/S0218196709005196 · Zbl 1182.16037 · doi:10.1142/S0218196709005196 [63] DOI: 10.1007/BF01232262 · Zbl 0738.20035 · doi:10.1007/BF01232262 [64] Wilson J., London Mathematical Society Monographs. New Series 19, in: Profinite Groups (1998) [65] DOI: 10.1002/mana.19811020107 · Zbl 0498.17011 · doi:10.1002/mana.19811020107 [66] DOI: 10.1002/mana.19901470109 · Zbl 0706.16010 · doi:10.1002/mana.19901470109 [67] DOI: 10.1007/978-1-4612-1380-2_7 · doi:10.1007/978-1-4612-1380-2_7 [68] DOI: 10.4134/JKMS.2007.44.5.1185 · Zbl 1171.16013 · doi:10.4134/JKMS.2007.44.5.1185
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.