## Nonabelian free subgroups in homomorphic images of valued quaternion division algebras

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- by Andrei S. Rapinchuk, Louis Rowen and Yoav Segev PDF
- Proc. Amer. Math. Soc.
**134**(2006), 3107-3114 Request permission

## Abstract:

Given a quaternion division algebra $D,$ a noncentral element $e \in D^\times$ is called*pure*if its square belongs to the center. A theorem of Rowen and Segev (2004) asserts that for any quaternion division algebra $D$ of positive characteristic $> 2$ and any pure element $e \in D^\times$ the quotient $D^{\times }/X(e)$ of $D^{\times }$ by the normal subgroup $X(e)$ generated by $e,$ is abelian-by-nilpotent-by-abelian. In this note we construct a quaternion division algebra $D$ of characteristic zero containing a pure element $e\in D$ such that $D^\times /X(e)$ contains a nonabelian free group. This demonstrates that the situation in characteristic zero is very different.

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## Additional Information

**Andrei S. Rapinchuk**- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
- MR Author ID: 206801
- Email: asr3x@unix.mail.virginia.edu
**Louis Rowen**- Affiliation: Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel
- MR Author ID: 151270
- Email: rowen@macs.biu.ac.il
**Yoav Segev**- Affiliation: Department of Mathematics, Ben-Gurion University, Beer-Sheva 84105, Israel
- MR Author ID: 225088
- Email: yoavs@math.bgu.ac.il
- Received by editor(s): March 3, 2005
- Received by editor(s) in revised form: May 14, 2005
- Published electronically: May 11, 2006
- Additional Notes: The first author was partially supported by BSF grant 2000-171, and by NSF grants DMS-0138315 and DMS-0502120.

The second author was partially supported by the Israel Science Foundation Center of Excellence.

The third author was partially supported by BSF grant 2000-171. - Communicated by: Jonathan I. Hall
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**134**(2006), 3107-3114 - MSC (2000): Primary 16K20, 16U60; Secondary 20G15, 12J20
- DOI: https://doi.org/10.1090/S0002-9939-06-08385-7
- MathSciNet review: 2231891