Free groups in quaternion algebras

Abstract

In Juriaans et al. (2009) [9] we constructed pairs of units u,v in Z-orders of a quaternion algebra over View the MathML source, d a positive and square free integer with View the MathML source, such that 〈un,vn〉 is free for some n∈N. Here we extend this result to any imaginary quadratic extension of Q, thus including matrix algebras. More precisely, we show that 〈un,vn〉 is a free group for all n⩾1 and d>2 and for d=2 and all n⩾2. The units we use arise from Pellʼs and Gaussʼ equations.