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

Abstract

We show that $\mathcal{U}(\mathbb{Z}G)$, the unit group of the integral group ring $\mathbb{Z} G$, 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. Crucial in the proof is the construction of amalgamated decompositions of the elementary group $\operatorname{E}_2(\mathcal{O})$, where $\mathcal{O}$ is an order in a rational division algebra. A major step is to introduce subgroups $\operatorname{E}_2(\Gamma_n(\mathbb{Z}))$ inside the so-called higher modular groups $\operatorname{SL}_+(\Gamma_n(\mathbb{Z}))$, which are discrete subgroups of certain $2 \times 2$ matrix groups with entries in a Clifford algebra. The groups $\operatorname{E}_2(\Gamma_n(\mathbb{Z}))$ mimic the elementary groups in linear groups over rings. We prove that $\operatorname{E}_2(\Gamma_n(\mathbb{Z}))$ has in general a non-trivial decomposition as a free product with amalgamated subgroup $\operatorname{E}_2(\Gamma_{n-1}(\mathbb{Z}))$. From this we obtain that also the higher modular groups do have a very clearly structured amalgam decompositions in low dimensions.