Abelianization and Fixed Point Properties of Units in Integral Group Rings

Let G be a finite group and 𝒰 (ℤ G) the unit group of the integral group ring ℤ G. We prove a unit theorem, namely a characterization of when 𝒰(ℤG) satisfies Kazhdan's property (T), both in terms of the finite group G and in terms of the simple components of the semisimple algebra ℚG. Furthermore, it is shown that for 𝒰( ℤ G) this property is equivalent to the weaker property FAb (i.e. every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property FA, denoted HFA. More precisely, it is described when all subgroups of finite index in 𝒰 (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 SL_n(𝒪), where 𝒪 is an order in a finite dimensional semisimple 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 ℚ G. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups E_n(D) of SL_n(D). These groups are well understood except in the degenerate case of lower rank, i.e. for SL_2(𝒪) with 𝒪 an order in a division algebra D with a finite number of units. In this setting we determine Serre's property for E_2(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-rank.