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Uniform nonamenability of subgroups of free Burnside groups of odd period. (English. Russian original) Zbl 1213.20036
Math. Notes 85, No. 4, 496-502 (2009); translation from Mat. Zametki 85, No. 4, 516-523 (2009).
Summary: A famous theorem of Adyan states that, for any \(m\geq 2\) and any odd \(n\geq 665\), the free \(m\)-generated Burnside group \(B(m,n)\) of period \(n\) is not amenable. It is proved in the present paper that every noncyclic subgroup of the free Burnside group \(B(m,n)\) of odd period \(n\geq 1003\) is a uniformly nonamenable group. This result implies the affirmative answer, for odd \(n\geq 1003\), to the following de la Harpe question: Is it true that the infinite free Burnside group \(B(m,n)\) has uniform exponential growth? It is also proved that every \(S\)-ball of radius \((400n)^3\) contains two elements which form a basis of a free periodic subgroup of rank 2 in \(B(m,n)\), where \(S\) is an arbitrary set of elements generating a noncyclic subgroup of \(B(m,n)\).

MSC:
20F50 Periodic groups; locally finite groups
20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
43A07 Means on groups, semigroups, etc.; amenable groups
20F06 Cancellation theory of groups; application of van Kampen diagrams
20F69 Asymptotic properties of groups
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[1] G. N. Arzhantseva, J. Burillo, M. Lustig, L. Reeves, H. Short, and E. Ventura, ”Uniform non-amenability,” Adv. Math. 197(2), 499–522 (2005). · Zbl 1077.43001 · doi:10.1016/j.aim.2004.10.013
[2] E. Føner, ”On groups with full Banach mean value,” Math. Scand. 3, 243–254 (1955). · Zbl 0067.01203 · doi:10.7146/math.scand.a-10442
[3] I. Namioka, ”Føner’s condition for amenable semi-groups,” Math. Scand 15, 18–28 (1964). · Zbl 0138.38001 · doi:10.7146/math.scand.a-10723
[4] A. Hulanicki, ”Means and Føner condition on locally compact groups,” Studia Math. 27, 87–104 (1966). · Zbl 0165.48701 · doi:10.4064/sm-27-2-87-104
[5] J. von Neumann, ”Zur allgemeinen Theorie des Masses,” Fundam. Math. 13, 73–116 (1929). · JFM 55.0151.01 · doi:10.4064/fm-13-1-73-116
[6] S. I. Adyan [Adjan], ”An axiomaticmethod for the construction of groups with prescribed properties,” Uspekhi Mat. Nauk 32(1), 3–15 (1977) [in Russian].
[7] S. I. Adyan, ”Random walks on free periodic groups,” Izv. Akad. Nauk SSSR Ser. Mat. 46(6), 1139–1149 (1982) [Math. USSR-Izv. 21 (3), 425–434 (1983)]. · Zbl 0512.60012
[8] I. G. Lysenok, ”Some algorithmic properties of hyperbolic groups,” Izv. Akad. Nauk SSSR Ser. Mat. 53(4), 814–832 (1989) [Math. USSR-Izv. 35 (1), 145–163 (1990)]. · Zbl 0692.20022
[9] P. de la Harpe and A. Valette, ”La propriété (T) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger),” in Astérisque (Soc. Math. France, Paris, 1989), Vol. 175. · Zbl 0759.22001
[10] D. V. Osin, ”Weakly amenable groups,” in Computational and Statistical Group Theory, Contemp. Math., Las Vegas, NV/Hoboken, NJ, 2001 (Amer. Math. Soc., Providence, RI, 2002), Vol. 298, pp. 105–113. · Zbl 1017.43001
[11] D. V. Osin, ”Uniform non-amenability of free Burnside groups,” Arch. Math. (Basel) 88(5), 403–412 (2007). · Zbl 1173.43002 · doi:10.1007/s00013-006-2002-5
[12] S. V. Ivanov and A. Yu. Ol’shanskii, ”Some applications of graded diagrams in combinatorial group theory,” in Groups, London Math. Soc. Lecture Note Ser., v. 2, Proc. Int. Conf., St. Andrews, UK 1989, 160 (Cambridge Univ. Press, Cambridge, 1991), pp. 258–308.
[13] Y. Shalom, ”Explicit Kazhdan constants for representations of semisimple and arithmetic groups,” Ann. Inst. Fourier (Grenoble) 50(3), 833–863 (2000). · Zbl 0966.22004 · doi:10.5802/aif.1775
[14] P. de la Harpe, ”Uniform growth in groups of exponential growth,” Geom. Dedicata 95(1), 1–17 (2002). · Zbl 1025.20027 · doi:10.1023/A:1021273024728
[15] S. I. Adyan [Adian], The Burnside Problem and Identities in Groups (Nauka, Moscow, 1975; Springer-Verlag, Berlin-New York, 1979).
[16] V. S. Atabekyan, ”On subgroups of free periodic groups of odd period n 1003,” Izv. Ross. Akad. Nauk Ser. Mat. (in press). · Zbl 1152.20034
[17] V. S. Atabekyan, ”Simple and free periodic groups,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. (6), 76–78 (1987) [Moscow Univ. Math. Bull. 42 (6), 80–82 (1987)]. · Zbl 0661.20027
[18] M. Koubi, ”Croissance uniforme dans les groupes hyperboliques,” Ann. Inst. Fourier (Grenoble) 48(5), 1441–1453 (1998). · Zbl 0914.20033 · doi:10.5802/aif.1661
[19] S. I. Adyan, ”The Burnside problem on periodic groups, and related problems,” Current problems in mathematics, 1 (Ross. Akad. Nauk, Inst. Mat. im. Steklova, Moscow, 2003), pp. 5–29 [in Russian]. · Zbl 1355.20024
[20] V. L. Shirvanyan [Širvanjan], ”Imbedding of the groupB(, n) in the groupB(2, n),” Izv. Akad.Nauk SSSR Ser. Mat. 40(1), 190–208 (1976) [in Russian]. · Zbl 0336.20027
[21] S. I. Adyan and I. G. Lysenok, ”Groups, all of whose proper subgroups are finite cyclic,” Izv. Akad. Nauk SSSR Ser. Mat. 55(5), 933–990 (1991) [Math. USSR-Izv. 39 (2), 905–957 (1992)].
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