Abelianization and fixed point properties of linear groups and units in integral group rings

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Abstract

Let 퐺 be a finite group and  (Z퐺) the unit group of the integral group ring Z퐺. We prove a unit theorem, namely a characterization of when  (Z퐺) satisfies Kazhdan’s property (T), both in terms of the finite group 퐺 and in terms of the simple compo- nents of the semisimple algebra Q퐺. Furthermore, it is shown that for  (Z퐺) this property is equivalent to the weaker property FAb (i.e. every subgroup of finite in- dex 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  (Z퐺) have both finite abelianization and are not a non-trivial amal- gamated product.
A crucial step for this is a reduction to arithmetic groups SL푛(), where  is an order in a finite dimensional semisimple Q-algebra 퐷, and finite groups 퐺 which have the so-called cut property. For such groups 퐺 we describe the simple epimorphic images of Q퐺. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups E푛(퐷) of SL푛(퐷). These groups are well understood except in the degenerate case of lower rank, i.e. for SL2() with  an order in a division algebra 퐷 with a finite number of units. In this setting we de- termine Serre’s property FA for E2() 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.
Original languageEnglish
Number of pages47
JournalMathematische Nachrichten
Publication statusAccepted/In press - 14 Jan 2021

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