GROUP ALGEBRAS OF KLEINIAN TYPE AND GROUPS OF UNITS
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 RGis virtually a direct product of free-by-free groups. The study of Kleinian groups goes back to the works of Poincar´ (Poi) and Bianchi (Bia) and it has been an active field of research ever since. In the last decades, it is 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 notion 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 a Z-order in A. Let R 1 denote the group of units of R of reduced norm 1. Every embedding � : K → C induces an embedding � : A → Md(C), where d is the degree of A. Furthermore, �(R 1 ) ⊆ SLd(C). We say that A is of Kleinian type if either A = K or A is a quaternion algebra over K and �(R 1 ) is a discrete subgroup of SL2(C) for some embeddingof K in C. More generally, an algebra of Kleinian type (PRR) is by definition a direct sum of simple algebras of Kleinian type. A finite group G is of Kleinian type if and only if the rational group algebra QG is of Kleinian type. The finite groups of Kleinian type have been classified in (JPRRZ) where it has been also proved that a finite group G is of Kleinian type if and only if the group of units ZG ∗ of its integral group ring ZG is commensurable with a direct product of free-by-free groups. Recently, Alan Reid asked us in a private communication about the consequences of replacing the ring of rational integers by another ring of integers. This leads to the following two problems: Problem 1. Classify the group algebras of Kleinian type of finite groups over number fields. Problem 2. Given a group algebra of Kleinian type KG, describe the structure of the group of units of the group ring RG for R an order in K.