Abelianization and fixed point properties of units in integral group rings

Abstract

Let $G$ be a finite group and $\mathcal{U} (\mathbb{Z} G)$ the unit group of the integral group ring $\mathbb{Z} G$. We prove a unit theorem, namely a characterization of when $\mathcal{U}(\mathbb{Z}G)$ satisfies Kazhdan's property $(\operatorname{T})$, both in terms of the finite group $G$ and in terms of the simple components of the semisimple algebra $\mathbb{Q}G$. Furthermore, it is shown that for $\mathcal{U}( \mathbb{Z} G)$ this property is equivalent to the weaker property $\operatorname{FAb}$ (i.e. every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property $\operatorname{FA}$, denoted $\operatorname{HFA}$. More precisely, it is described when all subgroups of finite index in $\mathcal{U} (\mathbb{Z} G)$ have both finite abelianization and are not a non-trivial amalgamated product. A crucial step for this is a reduction to arithmetic groups $\operatorname{SL}_n(\mathcal{O})$, where $\mathcal{O}$ is an order in a finite dimensional semisimple $\mathbb{Q}$-algebra $D$, and finite groups $G$ which have the so-called cut property. For such groups $G$ we describe the simple epimorphic images of $\mathbb{Q} G$. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups $\operatorname{E}_n(D)$ of $\operatorname{SL}_n(D)$. These groups are well understood except in the degenerate case of lower rank, i.e.\ for $\operatorname{SL}_2(\mathcal{O})$ with $\mathcal{O}$ an order in a division algebra $D$ with a finite number of units. In this setting we determine Serre's property \FA for $\operatorname{E}_2(\mathcal{O})$ and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its $\mathbb{Z}$-rank.