QUATERNION ALGEBRAS, ARITHMETIC KLEINIAN GROUPS AND Z-LATTICES

Let K be a quadratic extension of Q, B a quaternion alge-bra over Q and A = B ⊗Q K. Let O be a maximal order in A extending an order in B. The projective norm one group PO1 is shown to be isomorphic to the spinorial kernel group O′(L), for an explicitly determined quadratic Z-lattice L of rank four, in several general situations. In other cases, only the local structures of O and L are given at each prime. Both definite and indefinite lattices are covered. Some results for quadratic global field extensions K/F and maximal S-orders are given. There is a description of the F-quaternion subal-gebras of A, and also of their norm one groups as stabilizer subgroups and as unitary groups. Conjugacy classes of the Fuchsian subgroups of PO1 corresponding to stabilizer sub-groups are studied. 1. Introduction