Abelianization and fixed point properties of units in integral group rings

Abstract

Let G be a finite group and U(ZG)${\mathcal {U}}({\mathbb {Z}}G)$ the unit group of the integral group ring ZG${\mathbb {Z}}G$. We prove a unit theorem, namely, a characterization of when U(ZG)$\mathcal {U}(\mathbb {Z}G)$ satisfies Kazhdan's property (T)$(\operatorname{T})$, both in terms of the finite group G and in terms of the simple components of the semisimple algebra QG$\mathbb {Q}G$. Furthermore, it is shown that for U(ZG)$\mathcal {U}(\mathbb {Z} G)$, this property is equivalent to the weaker property FAb$\operatorname{FAb}$ (i.e., every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property FA$\operatorname{FA}$, denoted HFA$\operatorname{HFA}$. More precisely, it is described when all subgroups of finite index in U(ZG)${\mathcal {U}}({\mathbb {Z}}G)$ have both finite abelianization and are not a nontrivial amalgamated product. A crucial step for this is a reduction to arithmetic groups SLn(O)$\operatorname{SL}_n(\mathcal {O})$, where O$\mathcal {O}$ is an order in a finite-dimensional semisimple Q${\mathbb {Q}}$-algebra D, and finite groups G, which have the so-called cut property. For such groups G, we describe the simple epimorphic images of QG$\mathbb {Q} G$. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups En(D)$\operatorname{E}_n(D)$ of SLn(D)$\operatorname{SL}_n(D)$. These groups are well understood except in the degenerate case of lower rank, that is, for SL2(O)$\operatorname{SL}_2(\mathcal {O})$ with O$\mathcal {O}$ an order in a division algebra D with a finite number of units. In this setting, we determine Serre's property FA for E2(O)$\operatorname{E}_2(\mathcal {O})$ and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its Z$\mathbb {Z}$-rank.