Hyperbolic unit groups and quaternion algebras

Abstract

We classify the quadratic extensions K = Q[√ d] and the finite groups G for which the group ring o_K[G] of G over the ring o_K of integers of K has the property that the group U₁ (o_K[G]) of units of augmentation 1 is hyperbolic. We also construct units in the Z-order H(o_K) of the quaternion algebra H(K) = (-1,-1/K) , when it is a division algebra.