We restrict the types of 2 × 2-matrix rings which can occur as simple components in the Wedderburn decomposition of the rational group algebra of a finite group. This results in a description up to commensurability of the group of units of the integral group ring ZG for all finite groups G that do not have a non-commutative Frobenius complement as a quotient.

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring ZG of a finite nilpotent group G, this provided the rational group algebra QG does not have simple components that are division classical quaternion algebras or two-by-two matrices over a classical quaternion algebra with centre Q. The main difficulty is to deal with orders in quaternion algebras over the rationals or a quadratic imaginary extension of the rationals. In order to deal with these we give a finite and easy implementable algorithm to compute a fundamental domain in the hyperbolic three space H 3 (respectively hyperbolic two space H 2 ) for a discrete subgroup of PSL2(C) (respectively PSL2(R)) of finite covolume. Our results on group rings are a continuation of earlier work of Ritter and Sehgal, Jespers and Leal.

Fixed point properties and the abelianization of arithmetic subgroups Γ of SLn(D) and its elementary subgroup En(D) are well understood except in the degenerate case of lower rank, i.e. n = 2 and Γ = SL2(O) with O an order in a division algebra D with a finite number of units. In this setting we determine Serre's property FA for E2(O) and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its Z-rank. Thenceforth, we investigate applications in integral representation theory of finite groups. We obtain a characterization of when the unit group U(ZG) of the integral group ring ZG satisfies Kazhdan's property (T), both in terms of the finite group G and in terms of the simple components of the semisimple algebra QG. Furthermore, it is shown that for U(ZG) this property is equivalent to a hereditary version of property FA, denoted HFA, and even the significantly weaker property FAb (i.e. every subgroup of finite index has finite abelianization). A crucial step for this is a reduction to arithmetic groups SLn(O) and finite groups G which have the so-called cut property. For such groups G we describe the simple epimorphic images of QG. Contents 1 2 A. B ÄCHLE, G. JANSSENS, E. JESPERS, A. KIEFER, AND D. TEMMERMAN 5. Property FR and HFR for E 2 (O) 29 5.1. Property FR for the groups G R,K with applications to FR for E 2 (O) 30 5.2. Property FR for the Borel with a view on GE 2 (O) 34 Chapter III. Applications to U (ZG) 36 6. Exceptional components and cut groups 36 6.1. FA and cut groups 36 6.2. Higher rank and exceptional components 38 6.3. Exceptional components of cut groups 40 7. Property HFA 43 8. Property FA 47 Appendix A. Groups with faithful exceptional 2 × 2 components 51 Appendix B. Some Group Presentations 53 References 54

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The problem of describing the group of units U(ZG) of the integral group ring ZG of a finite group G has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order O in the simple epimorphic images A of the rational group algebra QG. Making use of the presentation part of Poincaré's Polyhedron Theorem, Pita, del Río and Ruiz proposed such a method for a large family of finite groups G and consequently Jespers, Pita, del Río, Ruiz and Zalesskii described the structure of U(ZG) for a large family of finite groups G. In order to handle many more groups, one would like to extend Poincaré's Method to discontinuous subgroups of the group of isometries of a direct product of hyperbolic spaces. If the algebra A has degree 2 then via the Galois embeddings of the centre of the algebra A one considers the group of reduced norm one elements of the order O as such a group and thus one would obtain a solution to the mentioned problem. This would provide presentations of the unit group of orders in the simple components of degree 2 of QG and in particular describe the unit group of ZG for every group G with irreducible character degrees less than or equal to 2. The aim of this paper is to initiate this approach by executing this method on the Hilbert modular group, i.e. the projective linear group of degree two over the ring of integers in a real quadratic extension of the rationals. This group acts discontinuously on a direct product of two hyperbolic spaces of dimension two. The fundamental domain constructed is an analogue of the Ford domain of a Fuchsian or a Kleinian group.