A dichotomy for integral group rings via higher modular groups as amalgamated products

Abstract

We show that U(ZG), the unit group of the integral group ring ZG, either satisfies Kazhdan's property (T) or is, up to commensurability, a non-trivial amalgamated product, in case G is a finite group satisfying some mild conditions. A key step in the proof is the construction of amalgamated decompositions of the elementary group E2(O), where O is an order in rational division algebra, and of certain arithmetic groups Γ. The methods for the latter turn out to work in much greater generality and most notably are carried out to obtain amalgam decompositions for the higher modular groups SL+(Γn(Z)), with n≤4, which can be seen as higher dimensional versions of modular and Bianchi groups. For this we introduce a subgroup mimicking the elementary linear group, denoted E2(Γn(Z)). We prove that E2(Γn(Z)) has always a non-trivial decomposition as a free product with amalgamated subgroup E2(Γn−1(Z)).