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Periodic products of groups. (English. Russian original) Zbl 1377.20026
J. Contemp. Math. Anal., Armen. Acad. Sci. 52, No. 3, 111-117 (2017); translation from Izv. Nats. Akad. Nauk Armen., Mat. 52, No. 3, 3-15 (2017).
Summary: In this paper, we provide an overview of the results relating to the \(n\)-periodic products of groups that have been obtained in recent years by the authors of the present paper, as well as some results obtained by other authors in this direction. The periodic products were introduced by the first author [Proc. Steklov Inst. Math. 142, 1–19 (1976; Zbl 0424.20020); translation from Tr. Mat. Inst. Steklov 142, 3–21 (1976)] to solve the Maltsev’s well-known problem. It was shown that the periodic products are exact, associative and hereditary for subgroups. They also possess some other important properties such as the Hopf property, the \(C*\)-simplicity, the uniform non-amenability, the \(SQ\)-universality, etc. It was proved that the \(n\)-periodic products of groups can uniquely be characterized by means of certain quite specific and simply formulated properties. These properties allow to extend to \(n\)-periodic products of various families of groups a number of results previously obtained for free periodic groups \(B(m,n)\). In particular, we describe the finite subgroups of \(n\)-periodic products, Also, we analyze and extend the simplicity criterion of \(n\)-periodic products obtained previously by the first author [Math. Notes 88, No. 6, 771–775 (2010; Zbl 1230.20025); translation from Mat. Zametki 88, No. 6, 803–810 (2010)].

20F50 Periodic groups; locally finite groups
20F05 Generators, relations, and presentations of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F28 Automorphism groups of groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
Full Text: DOI
[1] Adian, S. I., Periodic products of groups, Trudy MIAN SSSR, 142, 3-21, (1976)
[2] Adian, S. I., Once more on periodic products of groups and on a problem of A. I.maltsev, Mat. Zametki, 88, 803-810, (2010)
[3] S. I. Adian, The Burnside problem and identities in groups (Nauka, Moscow, 1975).
[4] Adian, S. I., New estimates of odd exponents of infinite Burnside groups, Trudy MIAN, 289, 41-82, (2015) · Zbl 1343.20040
[5] Adian, S. I., On the simplicity of periodic products of groups, Dokl. Akad. Nauk SSSR, 241, 745-748, (1978) · Zbl 0416.54023
[6] Adian, S. I.; Atabekyan, V. S., The Hopfian property of n-periodic products of groups, Mat. Zametki, 95, 483-491, (2014) · Zbl 1326.20040
[7] Adian, S. I., Random walks on free periodic groups, Izv. Akad. Nauk SSSR Ser. Mat., 46, 1139-1149, (1982) · Zbl 0512.60012
[8] Adian, S. I.; Atabekyan, V. S., Characteristic properties and uniform non-amenability of n-periodic products of groups, Izv.RAN. Ser. Mat., 79, 3-18, (2015) · Zbl 1360.20018
[9] S. I. Adian, V. S. Atabekyan, “On free groups of infinitely based varieties of S.I. Adian”, Izv. RAN.Ser. Mat., 81 (5), 2017. · Zbl 1436.20044
[10] Atabekyan, V. S., Uniform nonamenability of subgroups of free Burnside groups of odd period, Mat. Zametki, 85, 516-523, (2009) · Zbl 1213.20036
[11] Atabekyan, V. S., Monomorphisms of free Burnside groups, Mat. Zametki, 86, 483-490, (2009) · Zbl 1200.20028
[12] Atabekyan, V. S., On normal subgroups in the periodic products of S.I. Adian, TrudyMIAN, 274, 15-31, (2011)
[13] Ivanov, S. V., On subgroups of free Burnside groups of large odd exponent, Illinois J. Math., 47, 299-304, (2003) · Zbl 1036.20029
[14] A. Yu. Olshanskii, The Geometry of Defning Relations in Groups (Kluwer, Amsterdam, 1991).
[15] Harpe, P., On simplicity of reduced \(C\)*-algebras of groups, Bull. Lond. Math. Soc., 39, 1-26, (2007) · Zbl 1123.22004
[16] Breuillard, E.; Kalantar, M.; Kennedy, M.; Ozawa, N., \(C\)*-simplicity and the unique trace property for discrete groups, ArXiv:1410., 2518, 1-20, (2014)
[17] Adian, S. I.; Atabekyan, V. S., \(C\)*-simplicity of n-periodic products, Mat. Zametki, 99, 643-648, (2016)
[18] Olshanskii, A. Yu.; Osin, D. V., \(C\)*-simple groups without free subgroups, Groups Geom.Dyn., 8, 933-983, (2014) · Zbl 1354.22009
[19] V. S. Atabekyan, A.L. Gevorgyan, Sh.A. Stepanyan, “The uniqueness trace property of n-periodic products of groups”, Journal of ContemporaryMathematical Analysis, 52 (5), 2017. · Zbl 1378.20041
[20] Neshadim, M. V., Free products of groups have no outer normal automorphisms, Algebra i Logika, 35, 562-566, (1996)
[21] Atabekyan, V. S., Normal automorphisms of free Burnside groups, Izv.RAN. Ser.Mat., 75, 3-18, (2011) · Zbl 1227.20030
[22] Atabekyan, V. S., Non-f-admissible normal subgroups of free Burnside groups, Journal of Contemporary Mathematical Analysis, 45, 112-22, (2010) · Zbl 1299.20046
[23] Cherepanov, E. A., Normal automorphisms of free Burnside groups of large odd exponents, Internat. J. Algebra Comput, 16, 839-847, (2006) · Zbl 1115.20024
[24] Atabekyan, V. S.; Gevorgyan, A.L., On outer normal automorphisms of periodic products of groups, Journal of ContemporaryMathematical Analysis, 46, 289-92, (2011) · Zbl 1299.20047
[25] Gevorgyan, A. L., On automorphisms of periodic products of groups, Proceedings of the YSU, Physics and Mathematics, 2, 3-9, (2012) · Zbl 1309.20028
[26] Gevorgyan, A. L.; Stepanyan, Sh. A., On automorphisms of some periodic products of groups, Proceedings of the YSU, Physics and Mathematics, 2, 7-10, (2015) · Zbl 1322.20023
[27] Atabekyan, V. S., Splitting automorphisms of free Burnside groups, Sb. Math., 204, 182-189, (2013) · Zbl 1282.20025
[28] Atabekyan, V. S., Splitting automorphisms of order pk of free Burnside groups are inner, Math. Notes, 95, 586-589, (2014) · Zbl 1315.20030
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