Nonabelian free subgroups in homomorphic images of valued quaternion division algebras

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.