## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://www.math.psu.edu/era/.
**

Non-amenable finitely presented torsion-by-cyclic groups
**This journal is archived by the American Mathematical
Society. The master copy is available at
http://www.ams.org/era/
**

## Non-amenable finitely presented torsion-by-cyclic groups

### A. Yu. Ol'shanskii and M. V. Sapir

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

*Copyright 2001 American Mathematical Society*

**Retrieve entire article **

#### 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.

*Electronic Research Announcements of the AMS *Home page