Abelianization and fixed point properties of units in integral group rings
نویسندگان
چکیده
Let G be a finite group and U ( Z ) ${\mathcal {U}}({\mathbb {Z}}G)$ the unit of integral ring ${\mathbb {Z}}G$ . We prove theorem, namely, characterization when $\mathcal {U}(\mathbb {Z}G)$ satisfies Kazhdan's property T $(\operatorname{T})$ , both in terms simple components semisimple algebra Q $\mathbb {Q}G$ Furthermore, it is shown that for {Z} G)$ this equivalent to weaker FAb $\operatorname{FAb}$ (i.e., every subgroup index has abelianization), particular also hereditary version Serre's FA $\operatorname{FA}$ denoted HFA $\operatorname{HFA}$ More precisely, described all subgroups have abelianization are not nontrivial amalgamated product. A crucial step reduction arithmetic groups SL n O $\operatorname{SL}_n(\mathcal {O})$ where {O}$ an order finite-dimensional {Q}}$ -algebra D, G, which so-called cut property. For such we describe epimorphic images {Q} G$ The proof theorem fundamentally relies on fixed point properties elementary E D $\operatorname{E}_n(D)$ $\operatorname{SL}_n(D)$ These well understood except degenerate case lower rank, is, 2 $\operatorname{SL}_2(\mathcal with division number units. In setting, determine $\operatorname{E}_2(\mathcal its index. construct generic computable exact sequence describing abelianization, affording closed formula {Z}$ -rank.
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ژورنال
عنوان ژورنال: Mathematische Nachrichten
سال: 2022
ISSN: ['1522-2616', '0025-584X']
DOI: https://doi.org/10.1002/mana.202000514