 If formula_23 is any quaternion algebra over an imaginary quadratic number field formula_2 which is not isomorphic to a matrix algebra then the unit groups of orders in formula_23 are cocompact.The invariant trace field of a Kleinian group (or, through the monodromy image of the fundamental group, of an hyperbolic manifold) is the field generated by the traces of the squares of its elements.In the case of an arithmetic manifold whose fundamental groups is commensurable with that of a manifold derived from a quaternion algebra over a number field formula_2 the invariant trace field equals formula_2.