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

TL;DR

This paper establishes a dichotomy for the unit groups of integral group rings, showing they either have Kazhdan's property (T) or are non-trivially amalgamated products, using higher modular groups and amalgam decompositions.

Contribution

It introduces a novel approach using higher modular groups and amalgamated decompositions to analyze the structure of unit groups of integral group rings.

Findings

Unit groups either satisfy Kazhdan's property (T) or are amalgamated products.

Constructs amalgamated decompositions of elementary groups over orders in division algebras.

Shows that higher modular groups have structured amalgam decompositions in low dimensions.

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.