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

DOI:
https://doi.org/10.1090/S0002-9939-06-08385-7

Published electronically:
May 11, 2006

MathSciNet review:
2231891

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Abstract | References | Similar Articles | Additional Information

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

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