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We show that $\mathcal{U}(\mathbb{Z}G)$, the unit group of the integral groupring $\mathbb{Z} G$, either satisfies Kazhdan's property (T) or is, up tocommensurability, a non-trivial amalgamated product, in case $G$ is a finitegroup satisfying some mild conditions. Crucial in the proof is the constructionof amalgamated decompositions of the elementary group$\operatorname{E}_2(\mathcal{O})$, where $\mathcal{O}$ is an order in arational division algebra. A major step is to introduce subgroups$\operatorname{E}_2(\Gamma_n(\mathbb{Z}))$ inside the so-called higher modulargroups $\operatorname{SL}_+(\Gamma_n(\mathbb{Z}))$, which are discretesubgroups of certain $2 \times 2$ matrix groups with entries in a Cliffordalgebra. The groups $\operatorname{E}_2(\Gamma_n(\mathbb{Z}))$ mimic theelementary groups in linear groups over rings. We prove that$\operatorname{E}_2(\Gamma_n(\mathbb{Z}))$ has in general a non-trivialdecomposition as a free product with amalgamated subgroup$\operatorname{E}_2(\Gamma_{n-1}(\mathbb{Z}))$. From this we obtain that alsothe higher modular groups do have a very clearly structured amalgamdecompositions in low dimensions.

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