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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 $[x,y]^n=1$.

Copyright 2001 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 07 (2001), pp. 63-71
• Publisher Identifier: S 1079-6762(01)00095-6
• 2000 Mathematics Subject Classification. Primary 20F05, 43A07
• Key words and phrases. Amenable group, Burnside groups, free subgroups
• Received by the editors January 9, 2001
• Posted on July 3, 2001
• Communicated by Efim Zelmanov
• Comments (When Available)

A. Yu. Ol'shanskii

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, and Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

E-mail address: olsh@math.vanderbilt.edu, olshan@shabol.math.msu.su

M. V. Sapir

Department of Mathematics, Vanderbilt University, Nashville, TN 37240

E-mail address: msapir@math.vanderbilt.edu

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.