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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nonabelian free subgroups in homomorphic images of valued quaternion division algebras


Authors: Andrei S. Rapinchuk, Louis Rowen and Yoav Segev
Journal: Proc. Amer. Math. Soc. 134 (2006), 3107-3114
MSC (2000): Primary 16K20, 16U60; Secondary 20G15, 12J20
Published electronically: May 11, 2006
MathSciNet review: 2231891
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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
Email: asr3x@unix.mail.virginia.edu

Louis Rowen
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel
Email: rowen@macs.biu.ac.il

Yoav Segev
Affiliation: Department of Mathematics, Ben-Gurion University, Beer-Sheva 84105, Israel
Email: yoavs@math.bgu.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08385-7
PII: S 0002-9939(06)08385-7
Keywords: Quaternion division algebra, multiplicative group, valuation, residue algebra
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
Article copyright: © Copyright 2006 American Mathematical Society