GROUP ALGEBRAS OF KLEINIAN TYPE AND GROUPS OF UNITS

Abstract. The algebras of Kleinian type are finite dimensional semisimple rational algebras A such that the group of units of an order in A is commensurable with a direct product of Kleinian groups. We classify the Schur algebras of Kleinian type and the group algebras of Kleinian type. As an application, we characterize the group rings RG, with R an order in a number field and G a finite group, such that the group of units of RG is virtually a direct product of free-by-free groups. The study of Kleinian groups goes back to the works of Poincare ́ [Poi] and Bianchi [Bia] and it has been an active field of research ever since. During the last decades, Kleinian groups have been strongly related to the Geometrization Program of Thurston for the classification of 3-manifolds [EGM, MR, Mas, Thu]. The use of the methods of Kleinian groups to the study of the groups of units of group rings was started in [PRR] and led to the notions of algebras of Kleinian type and finite groups of Kleinian type. Let K be a number field, A a central simple K-algebra and R an order in A. (By order we mean always a Z-order.) Let R1 denote the group of units of R of reduced norm 1. Every embedding σ: K → C induces an embedding σ: A → Md(C), where d is the degree of A. Furthermore, σ(R1) ⊆ SLd(C). We say that A is of Kleinian type if either A = K or A is a quaternion algebra over K and σ(R1) is a discrete subgroup of SL2(C) for some embedding σ of K in C. More generally, an algebra of Kleinian type [PRR] is, by definition