Hyperbolic Unit Groups and Quaternion Algebras

We Classify the rational quadratic extensions K and the finite groups G for which the group ring R[G] of G over the ring R of integers of K has the property that the group of units of augmentation 1 of R[G] is hyperbolic. We also construct units in a non-split quaternion algebra over R.
Comment: 15 pages, this work is part of the PHd. Thesis of the third author. The paper was accepted in Proceedings Mathematical Sciences