Abelianization and fixed point properties of units in integral group rings
(2023) Mathematische Nachrichten — Vol. 296, n° 1, p. 8-56 (2023)
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- Let G be a finite group and U(ZG) the unit group of the integral group ring ZG. 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(ZG) 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(ZG) 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 QG. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups En(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 FA 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.
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Bächle, A., Janssens, G., Jespers, E., Kiefer, A., & Temmerman, D. (2023). Abelianization and fixed point properties of units in integral group rings. Mathematische Nachrichten, 296(1), 8-56. https://doi.org/10.1002/mana.202000514 (Original work published 2023)