Questions in algebra and mathematical logic. Scientific heritage of S. I. Adian

V. S. Atabekyan   ; L. D. Beklemishev   ; V. S. Guba   ; I. G. Lysenok   ; A. A. Razborov   ; A. L. Semenov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 1, pp. 1-27

Résumé

This is a survey of results on the Burnside problem and properties of Burnside groups, the finite basis problem for group identities, periodic products of groups and Malcev's problem, construction of groups with special properties (Tarski monsters), constructive bounds in the Burnside–Magnus problem, and algorithmic problems: the problem of recognition of group properties, the word problem for semigroups with one relation, and semi-Thue systems. The focus is on the most important results obtained in papers of Adian and his students. Bibliography: 81 titles.

Détail

Keywords: combinatorial theory of groups and semigroups, periodic groups, Burnside problem, finite basis problem, algorithmic problems, Adian–Rabin theorem.

Bibliographie

[1] P. S. Novikov, S. I. Adian, “Infinite periodic groups. I”, Math. USSR-Izv., 2:1 (1968), 209–236 ; II:2, 241–479 ; III:3, 665–685 | DOI | DOI | DOI | MR | MR | MR | Zbl

[2] P. S. Novikov, S. I. Adian, “Defining relations and the word problem for free periodic groups of odd order”, Math. USSR-Izv., 2:4 (1968), 935–942 | DOI | MR | Zbl

[3] S. I. Adian, The Burnside problem and identities in groups, Ergeb. Math. Grenzgeb., 95, Springer-Verlag, Berlin–New York, 1979, xi+311 pp. | MR | MR | Zbl | Zbl

[4] S. I. Adyan, “Algoritmicheskaya nerazreshimost problem raspoznavaniya nekotorykh svoistv grupp”, Dokl. AN SSSR, 103:4 (1955), 533–535 | MR | Zbl

[5] A. Markov, “Nerazreshimost problemy gomeomorfii”, Dokl. AN SSSR, 121:2 (1958), 218–220 | MR | Zbl

[6] L. D. Beklemishev, I. G. Lysenok, A. A. Mal'tsev, S. P. Novikov, M. R. Pentus, A. A. Razborov, A. L. Semenov, V. A. Uspenskii, “Sergei Ivanovich Adian (on his 75th birthday)”, Russian Math. Surveys, 61:3 (2006), 575–588 ; АлгоритмичРμСЃРєРёРμ РІРѕРїСЂРѕСЃС‹ алгРμР±СЂС‹ Рё логики, РЎР±РѕСЂРЅРёРє статРμРNo. Рљ 80-Р»Рμтию СЃРѕ РґРЅСЏ рождРμРЅРёСЏ акадРμРјРёРєР° РЎРμСЂРіРμСЏ Р�вановича РђРґСЏРЅР°, РўСЂ. РњР�РђРќ, 274, РњРђР�Рљ “Наука/Р�РЅС‚РμСЂРїРμСЂРёРѕРґРёРєР°”, Рњ., 2011, 10–14 | DOI | DOI | MR | Zbl | MR | Zbl

[7] P. S. Novikov, S. I. Adian, “On a semicontinuous function”, Proc. Steklov Inst. Math., 274 (2011), 4–8 | DOI | MR | MR | Zbl | Zbl

[8] V. G. Kanovei, V. A. Lyubetskii, Sovremennaya teoriya mnozhestv: borelevskie i proektivnye mnozhestva, MTsNMO, M., 2010, 320 pp.

[9] P. S. Novikov, “On the algorithmic insolvability of the word problem in group theory”, Amer. Math. Soc. Transl. Ser. 2, 9, Amer. Math. Soc., Providence, RI, 1958, 1–122 | DOI | MR | MR | Zbl | Zbl

[10] A. A. Markov, “Nevozmozhnost algorifmov raspoznavaniya nekotorykh svoistv assotsiativnykh sistem”, Dokl. AN SSSR, 77:6 (1951), 953–956 | MR | Zbl

[11] H. G. Rice, “Classes of recursively enumerable sets and their decision problems”, Trans. Amer. Math. Soc., 74:2 (1953), 358–366 | DOI | MR | Zbl

[12] V. A. Uspenskii, “Sistemy perechislimykh mnozhestv i ikh numeratsii”, Dokl. AN SSSR, 105:6 (1955), 1155–1158 | MR | Zbl

[13] S. I. Adyan, “Nerazreshimost nekotorykh algoritmicheskikh problem teorii grupp”, Tr. MMO, 6, GITTL, M., 1957, 231–298 | MR | Zbl

[14] S. I. Adyan, “Konechno-opredelennye gruppy i algoritmy”, Dokl. AN SSSR, 117:1 (1957), 9–12 | MR | Zbl

[15] M. O. Rabin, “Recursive unsolvability of group theoretic problems”, Ann. of Math. (2), 67 (1958), 172–194 | DOI | MR | Zbl

[16] G. Higman, “Subgroups of finitely presented groups”, Proc. Roy. Soc. London Ser. A, 262:1311 (1961), 455–475 | DOI | MR | Zbl

[17] C. F. Miller III, P. E. Schupp, “Embeddings into Hopfian groups”, J. Algebra, 17:2 (1971), 171–176 | DOI | MR | Zbl

[18] M. A. Shtan'ko, “Markov's theorem and algorithmically non-recognizable combinatorial manifolds”, Izv. Math., 68:1 (2004), 205–221 | DOI | DOI | MR | Zbl

[19] M. A. Stan'ko, “On the Markov theorem on algorithmic nonrecognizability of manifolds”, J. Math. Sci. (N. Y.), 146:1 (2007), 5622–5623 | DOI | MR | Zbl

[20] I. N. Sanov, “Reshenie problemy Bernsaida dlya pokazatelya 4”, Uch. zap. LGU. Ser. matem., 1940, no. 10, 166–170 | MR | Zbl

[21] M. Hall, Jr., “Solution of the Burnside problem for exponent six”, Illinois J. Math., 2:4B (1958), 764–786 | DOI | MR | Zbl

[22] V. A. Tartakovskii, “Reshenie problemy tozhdestva dlya grupp s $k$-sokratimym bazisom pri $k>6$”, Izv. AN SSSR. Ser. matem., 13:6 (1949), 483–494 | MR | Zbl

[23] S. E. Arshon, “Dokazatelstvo suschestvovaniya $n$-znachnykh beskonechnykh asimmetrichnykh posledovatelnostei”, Matem. sb., 2(44):4 (1937), 769–779 | Zbl

[24] P. S. Novikov, “On periodic groups”, Amer. Math. Soc. Transl. Ser. 2, 45, Amer. Math. Soc., Providence, RI, 1965, 19–22 | DOI | MR | Zbl

[25] E. S. Golod, “On nil-algebras and finitely approximable $p$-groups”, Amer. Math. Soc. Transl. Ser. 2, 48, Amer. Math. Soc., Providence, RI, 1965, 103–106 | DOI | MR | Zbl

[26] T. Delzant, M. Gromov, “Courbure mésoscopique et théorie de la toute petite simplification”, J. Topol., 1:4 (2008), 804–836 | DOI | MR | Zbl

[27] R. Coulon, “On the geometry of Burnside quotients of torsion free hyperbolic groups”, Internat. J. Algebra Comput., 24:3 (2014), 251–345 | DOI | MR | Zbl

[28] A. Yu. Ol'shanskiĭ, “On the Novikov–Adyan theorem”, Math. USSR-Sb., 46:2 (1983), 203–236 | DOI | MR | Zbl

[29] P. S. Novikov, S. I. Adian, “On abelian subgroups and the conjugacy problem in free periodic groups of odd order”, Math. USSR-Izv., 2:5 (1968), 1131–1144 | DOI | MR | Zbl

[30] S. I. Adjan, “On some torsion-free groups”, Math. USSR-Izv., 5:3 (1971), 475–484 | DOI | MR | Zbl

[31] S. I. Adjan, “The subgroups of free periodic groups of odd exponent”, Proc. Steklov Inst. Math., 112 (1971), 61–69 | MR | Zbl

[32] S. I. Adjan, “Infinite irreducible systems of group identities”, Math. USSR-Izv., 4:4 (1970), 721–739 | DOI | MR | Zbl

[33] M. Gromov, “Hyperbolic groups”, Essays in group theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75–263 | DOI | MR | Zbl

[34] A. Yu. Ol'shanskiĭ, “Groups of bounded period with subgroups of prime order”, Algebra and Logic, 21:5 (1982), 369–418 | DOI | MR | Zbl

[35] S. V. Ivanov, “The free Burnside groups of sufficiently large exponents”, Internat. J. Algebra Comput., 4:1-2 (1994), 1–308 | DOI | MR | Zbl

[36] I. G. Lysenok, “Infinite Burnside groups of even exponent”, Izv. Math., 60:3 (1996), 453–654 | DOI | DOI | MR | Zbl

[37] S. V. Ivanov, A. Yu. Ol'shanskii, “Hyperbolic groups and their quotients of bounded exponents”, Trans. Amer. Math. Soc., 348:6 (1996), 2091–2138 | DOI | MR | Zbl

[38] B. H. Neumann, “Identical relations in groups. I”, Math. Ann., 114:1 (1937), 506–525 | DOI | MR | Zbl

[39] H. Neumann, Varieties of groups, Ergeb. Math. Grenzgeb., 37, Springer-Verlag New York, Inc., New York, 1967, x+192 pp. | MR | MR | Zbl

[40] B. H. Neumann, “Varieties of groups”, Bull. Amer. Math. Soc., 73:5 (1967), 603–613 | DOI | MR | Zbl

[41] S. I. Adian, “The Burnside problem and related topics”, Russian Math. Surveys, 65:5 (2010), 805–855 | DOI | DOI | MR | Zbl

[42] S. I. Adjan, “Infinite irreducible systems of group identities”, Soviet Math. Dokl., 11 (1970), 113–115 | MR | Zbl

[43] A. J. Ol'šanskiĭ, “On the problem of a finite basis of identities in groups”, Math. USSR-Izv., 4:2 (1970), 381–389 | DOI | MR | Zbl

[44] M. R. Vaughan-Lee, “Uncountably many varieties of groups”, Bull. London Math. Soc., 2:3 (1970), 280–286 | DOI | MR | Zbl

[45] S. I. Adyan, “Periodic products of groups”, Proc. Steklov Inst. Math., 142 (1979), 1–19 | MR | Zbl

[46] A. G. Kurosh, Teoriya grupp, 3-e izd., Nauka, M., 1967, 648 pp. ; A. G. Kurosch, Gruppentheorie, überarb. erw. Aufl., v. I,\;II, Math. Lehrbucher Monogr. I. Abt. Math. Lehrbucher, v. I, III/I, Akademie-Verlag, Berlin, 1970, xxii+360 pp. ; v. II, III/II, 1972, xiv+358 pp. | MR | Zbl | MR | Zbl | MR | Zbl

[47] S. I. Adyan, “Once more on periodic products of groups and on a problem of A. I. Mal'tsev”, Math. Notes, 88:6 (2010), 771–775 | DOI | DOI | MR | Zbl

[48] S. I. Adyan, “On the simplicity of periodic products of groups”, Soviet Math. Dokl., 19:4 (1978), 910–913 | MR | Zbl

[49] S. I. Adian, V. S. Atabekyan, “The Hopfian property of $n$-periodic products of groups”, Math. Notes, 95:4 (2014), 443–449 | DOI | DOI | MR | Zbl

[50] S. I. Adian, V. S. Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110 | DOI | DOI | MR | Zbl

[51] S. I. Adyan, V. S. Atabekyan, “$C^*$-simplicity of $n$-periodic products”, Math. Notes, 99:5 (2016), 631–635 | DOI | DOI | MR | Zbl

[52] S. I. Adian, V. S. Atabekyan, “Central extensions of free periodic groups”, Sb. Math., 209:12 (2018), 1677–1689 | DOI | DOI | MR | Zbl

[53] S. I. Adian, V. S. Atabekyan, “On free groups in the infinitely based varieties of S. I. Adian”, Izv. Math., 81:5 (2017), 889–900 | DOI | DOI | MR | Zbl

[54] S. I. Adian, V. S. Atabekyan, “Normal automorphisms of free groups of infinitely based varieties”, Math. Notes, 108:2 (2020), 149–154 | DOI | DOI | Zbl

[55] S. I. Adian, V. S. Atabekyan, “$n$-torsion groups”, J. Contemp. Math. Anal., Armen. Acad. Sci., 54:6 (2019), 319–327 | DOI | Zbl

[56] V. S. Atabekyan, C. V. Ivanov, Dva zamechaniya o gruppakh ogranichennogo perioda, Dep. v VINITI 30.03.1987, No 2243-V87, 1987, 23 pp.

[57] S. I. Adyan, I. G. Lysenok, “On groups all of whose proper subgroups are finite cyclic”, Math. USSR-Izv., 39:2 (1992), 905–957 | DOI | MR | Zbl

[58] W. Magnus, “A connection between the Baker–Hausdorff formula and a problem of Burnside”, Ann. of Math. (2), 52 (1950), 111–126 | DOI | MR | Zbl

[59] E. I. Zel'manov, “Solution of the restricted Burnside problem for groups of odd exponent”, Math. USSR-Izv., 36:1 (1991), 41–60 | DOI | MR | Zbl

[60] E. I. Zel'manov, “A solution of the restricted Burnside problem for 2-groups”, Math. USSR-Sb., 72:2 (1992), 543–565 | DOI | MR | Zbl

[61] H. Zassenhaus, “Über Lie'sche Ringe mit Primzahlcharakteristik”, Abh. Math. Sem. Univ. Hamburg, 13:1 (1939), 1–100 | DOI | MR | Zbl

[62] A. I. Kostrikin, Around Burnside, Ergeb. Math. Grenzgeb. (3), 20, Springer-Verlag, Berlin, 1990, xii+220 pp. | DOI | MR | MR | Zbl | Zbl

[63] S. I. Adian, A. A. Razborov, “Periodic groups and Lie algebras”, Russian Math. Surveys, 42:2 (1987), 1–81 | DOI | MR | Zbl

[64] S. I. Adyan, N. N. Repin, “Exponential lower estimate of the degree of nilpotency of Engel Lie algebras”, Math. Notes, 39:3 (1986), 244–249 | DOI | MR | Zbl

[65] G. E. Wall, “On the Lie ring of a group of prime exponent”, Proceedings of the second international conference on the theory of groups (Australian Nat. Univ., Canberra, 1973), Lecture Notes in Math., 372, Springer, Berlin, 1974, 667–690 | DOI | MR | Zbl

[66] S. I. Adyan, N. N. Repin, “Lower estimates of the order of maximal periodic groups of prime period”, Math. Notes, 44:2 (1988), 571–579 | DOI | MR | Zbl

[67] M. Vaughan-Lee, E. I. Zelmanov, “Upper bounds in the restricted Burnside problem”, J. Algebra, 162:1 (1993), 107–145 | DOI | MR | Zbl

[68] H. E. Rose, Subrecursion: functions and hierarchies, Oxford Logic Guides, 9, The Clarendon Press, Oxford Univ. Press, New York, 1984, xiii+191 pp. | MR | Zbl

[69] W. Magnus, “Das Identitätsproblem für Gruppen mit einer definierenden Relation”, Math. Ann., 106:1 (1932), 295–307 | DOI | MR | Zbl

[70] S. I. Adjan, “Defining relations and algorithmic problems for groups and semigroups”, Proc. Steklov Inst. Math., 85 (1966), 1–152 | MR | Zbl

[71] S. I. Adjan, G. U. Oganesyan, “On the word and divisibility problems in semigroups with a single defining relation”, Math. USSR-Izv., 12:2 (1978), 207–212 | DOI | MR | Zbl

[72] S. I. Adyan, G. U. Oganesyan, “Problems of equality and divisibility in semigroups with a single defining relation”, Math. Notes, 41:3 (1987), 235–240 | DOI | MR | Zbl

[73] S. I. Adyan, “Transformations of words in a semigroup presented by a system of defining relations”, Algebra and Logic, 15:6 (1976), 379–386 | DOI | MR | Zbl

[74] Yu. V. Matiyasevich, “Simple examples of undecidable associative calculi”, Soviet Math. Dokl., 8 (1967), 555–557 | MR | Zbl

[75] Terese, Term rewriting systems, Cambridge Tracts Theoret. Comput. Sci., 55, eds. M. Bezem, J. W. Klop, R. de Vrijer, Cambridge Univ. Press, Cambridge, 2003, xxii+884 pp. | MR | Zbl

[76] S. I. Adian, “On a method for proving exact bounds on derivational complexity in Thue systems”, Math. Notes, 92:1 (2012), 3–15 | DOI | DOI | MR | Zbl

[77] Y. Matiyasevich, G. Sénizergues, “Decision problems for semi-Thue systems with a few rules”, Theoret. Comput. Sci., 330:1 (2005), 145–169 | DOI | MR | Zbl

[78] D. Hofbauer, J. Waldmann, “Termination of $\{aa\to bc, bb\to ac, cc\to ab\}$”, Inform. Process. Lett., 98:4 (2006), 156–158 | DOI | MR | Zbl

[79] S. I. Adian, I. G. Lysionok, J. G. Mennicke, “Defining relations and the algebraic structure of the group $\mathrm{SL}_2$ over integral Hamilton quaternions”, Internat. J. Algebra Comput., 7:1 (1997), 1–24 | DOI | MR | Zbl

[80] S. I. Adian, J. Mennicke, “On bounded generation of $\mathrm{SL}_n(\mathbf{Z})$”, Internat. J. Algebra Comput., 2:4 (1992), 357–365 | DOI | MR | Zbl

[81] S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71 | DOI | DOI | MR | Zbl