Electronic Research Announcements

Author(s): A. Yu. Ol'shanskii; M. V. Sapir
Journal: Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 63-71.
MSC (2000): Primary 20F05, 43A07
Posted: July 3, 2001
Retrieve article in: PDF DVI PostScript

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

1.: S. I. Adian. Random walks on free periodic groups. Izv. Akad. Nauk SSSR, Ser. Mat. 46 (1982), 1139-1149. MR 84m:43001
2.: S. I. Adian. Periodic products of groups. Number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov. 142 (1976), 3-21, 268. MR 80m:20030
3.: S. Banach, A. Tarski. Sur la décomposition des ensembles de points en parties respectivement congruentes. Fund. Math 6 (1924), 244-277.
4.: 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/msapir/publications.html.
5.: M. G. Brin and C. C. Squier. Groups of piecewise linear homeomorphisms of the real line. Invent. Math. 79 (1985), 485-498. MR 86h:57033
6.: J. W. Cannon, W. J. Floyd and W. R. Parry. Introductory notes on Richard Thompson's groups. L'Enseignement Mathématique (2) 42 (1996), 215-256. MR 98g:20058
7.: 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 24:A2627
8.: J. M. Cohen. Cogrowth and amenability of discrete groups. J. Funct. Anal. 48 (1982), no. 3, 301-309. MR 85e:43004
9.: Mahlon M. Day. Amenable semigroups. Illinois J. Math. 1 (1957), 509-544. MR 19:1067c
10.: Open problems in infinite-dimensional topology. Edited by Ross Geoghegan. The Proceedings of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979). Topology Proc. 4 (1979), no. 1, 287-338 (1980). MR 82a:57015
11.: F. P. Greenleaf. Invariant means on topological groups and their applications. Van Nostrand Reinhold, New York, 1969. MR 40:4776
12.: R. I. Grigorchuk. Symmetrical random walks on discrete groups. Multicomponent random systems. Adv. Probab. Related Topics, 6, Dekker, New York, 1980, 285-325. MR 83k:60016
13.: R. I. Grigorchuk. An example of a finitely presented amenable group that does not belong to the class EG. Mat. Sb. 189 (1) (1998), 79-100. MR 99b:20055
14.: F. Hausdorff. Grundzüge der Mengenlehre. Leipzig, 1914.
15.: S. V. Ivanov and A. Y. Ol'shanskii. Hyperbolic groups and their quotients of bounded exponents. Trans. Amer. Math. Soc. 348 (1996), no. 6, 2091-2138. MR 96m:20057
16.: Harry Kesten. Full Banach mean values on countable groups. Math. Scand. 7 (1959), 146-156. MR 22:2911
17.: O. G. Kharlampovich and M. V. Sapir. Algorithmic problems in varieties. Internat. J. Algebra Comput. 5 (1995), no. 4-5, 379-602. MR 96m:20045
18.: Kourovka Notebook. Unsolved Problems in Group Theory. 8th edition, Novosibirsk, 1982. MR 84m:20002
19.: Roger Lyndon and Paul Schupp. Combinatorial group theory. Springer-Verlag, 1977. MR 58:28182
20.: J. von Neumann. Zur allgemeinen Theorie des Masses. Fund. Math. 13 (1929), 73-116.
21.: P. S. Novikov and S. I. Adian. Infinite periodic groups. I, II, III, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 212-244, 251-524, 709-731. MR 39:1532a; MR 39:1532b; MR 39:1532c
22.: A. Yu. Ol'shanskii. An infinite simple torsion-free Noetherian group. Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 6, 1328-1393. MR 81i:20033
23.: A. Yu. Ol'shanskii. An infinite group with subgroups of prime order. Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 2, 309-321. MR 82a:20035
24.: A. Yu. Ol'shanskii. On the question of the existence of an invariant mean on a group. (Russian) Uspekhi Mat. Nauk 35 (1980), no. 4 (214), 199-200. MR 82b:43002
25.: A. Yu. Ol'shanskii. The geometry of defining relations in groups, Nauka, Moscow, 1989. MR 91i:20035
26.: A. Yu. Ol'shanskii. The SQ-universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119-132. MR 97b:20057
27.: A. Yu. Ol'shanskii, M. V. Sapir. Embeddings of relatively free groups into finitely presented groups. Contemporary Mathematics, 264 (2000), 23-47. CMP 2001:05
28.: A. Yu. Ol'shanskii and M. V. Sapir. Length and area functions on groups and quasi-isometric Higman embeddings. To appear, IJAC, 2000.
29.: A. Yu. Ol'shanskii and M. V. Sapir. Non-amenable finitely presented torsion-by-cyclic groups. (Submitted).
30.: 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/msapir/publications.html.
31.: W. Specht. Zur Theorie der messbaren Gruppen. Math. Z. 74 (1960), 325-366. MR 25:4058
32.: J. Tits. Free subgroups in linear groups. J. Algebra 20 (1972), 250-270. MR 44:4105

Retrieve articles in Electronic Research Announcements with MSC (2000): 20F05, 43A07

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

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

DOI: 10.1090/S1079-6762-01-00095-6
PII: S 1079-6762(01)00095-6
Keywords: Amenable group, Burnside groups, free subgroups
Received by editor(s): January 9, 2001
Posted: 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
Copyright of article: Copyright 2001, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1079-6762

Previous volume \| Table of contents \| Next volume Articles in press \| Most recent volume \| All volumes Previous Article \| Next Article

Currently published by the American Institute of Mathematical Sciences as Electronic Research Announcements in Mathematical Sciences


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS