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Non-amenable finitely presented torsion-by-cyclic groups


Authors: A. Yu. Ol’shanskii and M. V. Sapir
Journal: Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 63-71
MSC (2000): Primary 20F05, 43A07
DOI: https://doi.org/10.1090/S1079-6762-01-00095-6
Published electronically: July 3, 2001
MathSciNet review: 1852901
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent $n\gg 1$ by a cyclic group, so it satisfies the identity $[x,y]^n=1$.


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  • S. I. Adyan, Random walks on free periodic groups, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1139–1149, 1343 (Russian). MR 682486
  • S. I. Adjan, Periodic products of groups, Trudy Mat. Inst. Steklov. 142 (1976), 3–21, 268 (Russian). Number theory, mathematical analysis and their applications. MR 532668
  • BT S. Banach, A. Tarski. Sur la décomposition des ensembles de points en parties respectivement congruentes. Fund. Math 6 (1924), 244–277. BORS J. C. Birget, A. Yu. Ol’shanskii, E. Rips, M. V. Sapir. Isoperimetric functions of groups and computational complexity of the word problem, 1998 (submitted to Annals of Mathematics), preprint available at http://www.math.vanderbilt.edu/$\sim$msapir/publications.html.
  • Matthew G. Brin and Craig C. Squier, Groups of piecewise linear homeomorphisms of the real line, Invent. Math. 79 (1985), no. 3, 485–498. MR 782231, DOI 10.1007/BF01388519
  • J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215–256. MR 1426438
  • A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
  • Joel M. Cohen, Cogrowth and amenability of discrete groups, J. Funct. Anal. 48 (1982), no. 3, 301–309. MR 678175, DOI 10.1016/0022-1236(82)90090-8
  • Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
  • Open problems in infinite-dimensional topology, Topology Proc. 4 (1979), no. 1, 287–338 (1980). MR 583711
  • Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto-London, 1969. MR 0251549
  • R. I. Grigorchuk, Symmetrical random walks on discrete groups, Multicomponent random systems, Adv. Probab. Related Topics, vol. 6, Dekker, New York, 1980, pp. 285–325. MR 599539
  • R. I. Grigorchuk, An example of a finitely presented amenable group that does not belong to the class EG, Mat. Sb. 189 (1998), no. 1, 79–100 (Russian, with Russian summary); English transl., Sb. Math. 189 (1998), no. 1-2, 75–95. MR 1616436, DOI 10.1070/SM1998v189n01ABEH000293
  • Haus F. Hausdorff. Grundzüge der Mengenlehre. Leipzig, 1914.
  • S. V. Ivanov and A. Yu. Ol′shanskiĭ, Hyperbolic groups and their quotients of bounded exponents, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2091–2138. MR 1327257, DOI 10.1090/S0002-9947-96-01510-3
  • Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. MR 112053, DOI 10.7146/math.scand.a-10568
  • O. G. Kharlampovich and M. V. Sapir, Algorithmic problems in varieties, Internat. J. Algebra Comput. 5 (1995), no. 4-5, 379–602. MR 1361261, DOI 10.1142/S0218196795000227
  • V. D. Mazurov, Yu. I. Merzlyakov, and V. A. Churkin (eds.), Kourovskaya tetrad′, 8th ed., Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1982 (Russian). Nereshennye voprosy teorii grupp. [Unsolved problems in group theory]; Including problems from the 8th All-Union symposium on group theory held at Sumy, 1982. MR 725169
  • Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
  • vN J. von Neumann. Zur allgemeinen Theorie des Masses. Fund. Math. 13 (1929), 73–116.
  • L. C. Young, On an inequality of Marcel Riesz, Ann. of Math. (2) 40 (1939), 567–574. MR 39, DOI 10.2307/1968941
  • A. Ju. Ol′šanskiĭ, An infinite simple torsion-free Noetherian group, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 6, 1328–1393 (Russian). MR 567039
  • A. Ju. Ol′šanskiĭ, An infinite group with subgroups of prime orders, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 2, 309–321, 479 (Russian). MR 571100
  • A. Ju. Ol′šanskiĭ, On the question of the existence of an invariant mean on a group, Uspekhi Mat. Nauk 35 (1980), no. 4(214), 199–200 (Russian). MR 586204
  • A. Yu. Ol′shanskiĭ, Geometriya opredelyayushchikh sootnosheniĭ v gruppakh, Sovremennaya Algebra. [Modern Algebra], “Nauka”, Moscow, 1989 (Russian). With an English summary. MR 1024791
  • A. Yu. Ol′shanskiĭ, $\textrm {SQ}$-universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119–132 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 8, 1199–1211. MR 1357360, DOI 10.1070/SM1995v186n08ABEH000063
  • OS A. Yu. Ol’shanskii, M. V. Sapir. Embeddings of relatively free groups into finitely presented groups. Contemporary Mathematics, 264 (2000), 23–47. talk A. Yu. Ol’shanskii and M. V. Sapir. Length and area functions on groups and quasi-isometric Higman embeddings. To appear, IJAC, 2000. OSamen A. Yu. Ol’shanskii and M. V. Sapir. Non-amenable finitely presented torsion-by-cyclic groups. (Submitted). SBR M. V. Sapir, J. C. Birget, E. Rips. Isoperimetric and isodiametric functions of groups, 1997, submitted to Annals of Mathematics, preprint available at http://www.math.vanderbilt.edu/$\sim$msapir/publications.html.
  • Wilhelm Specht, Zur Theorie der messbaren Gruppen, Math. Z. 74 (1960), 325–366 (German). MR 140641, DOI 10.1007/BF01180494
  • J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 286898, DOI 10.1016/0021-8693(72)90058-0

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Additional Information

A. Yu. Ol’shanskii
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, TN 37240, and Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia
MR Author ID: 196218
Email: olsh@math.vanderbilt.edu, olshan@shabol.math.msu.su

M. V. Sapir
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, TN 37240
MR Author ID: 189574
Email: msapir@math.vanderbilt.edu

Keywords: Amenable group, Burnside groups, free subgroups
Received by editor(s): January 9, 2001
Published electronically: July 3, 2001
Additional Notes: Both authors were supported in part by the NSF grant DMS 0072307. In addition, the research of the first author was supported in part by the Russian fund for fundamental research 99-01-00894, and the research of the second author was supported in part by the NSF grant DMS 9978802.
Communicated by: Efim Zelmanov
Article copyright: © Copyright 2001 American Mathematical Society