Meumertzheim, Fabian: Embeddings of groups rings and L2-invariants. - Bonn, 2021. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
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author = {{Fabian Meumertzheim}},
title = {Embeddings of groups rings and L2-invariants},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2021,
month = jun,

note = {The strong Atiyah conjecture for a group G, which restricts the possible values taken by L2-Betti numbers of G-CW-complexes, implies far-reaching structural statements about a canonical *-regular ring RℚG, in which the group ring ℚG embeds: For a general group G with a uniform bound on the order of its finite subgroups this ring is semisimple and, in the torsion-free case, even a division ring. In this way, the conjecture, which readily generalizes to other subfields K of the complex numbers, yields answers to classical algebraic questions on the embeddability of group rings into division rings via analytic methods.
This thesis studies embeddings of group rings that are related to the Atiyah conjecture, with a special focus on the structure of the rings RKG. It begins with a review of important ring-theoretic notions in the first chapter and a detailed account of the current state of knowledge on the strong Atiyah conjecture in the second chapter.
The third chapter deals with the projective class group K0(RKG), with a particular emphasis on the effects of a change of the coefficient field K. As an application of our results, we show that for a sofic group the center-valued Atiyah conjecture over all fields containing sufficiently many roots of unity follows from that over the algebraic closure of ℚ. Furthermore, we show that for a sofic group G the ring RKG is unit-regular not only for uncountable but also for certain countable coefficient fields.
In the fourth chapter, we analyze to which extent the special properties of the ring RKG are relevant to the construction of the L2-invariants, where the latter include not only the L2-Betti numbers but also the universal L2-torsion and the L2-polytope introduced by Friedl and Lück. For this purpose, starting with an arbitrary ring homomorphism from the group ring ℤG of a torsion-free group G to a division ring, we construct fully algebraic analogues of these invariants, the so-called agrarian invariants. These invariants are then applied to two-generator one-relator groups, for which classical embedding results have long been known whereas the strong Atiyah conjecture has only very recently been proved for these groups. As a result, we are able to confirm the well-definedness of a polytope-valued invariant of these groups going back to Friedl and Tillmann. Furthermore, we can relate the diameter of the polytope to the complexity of possible HNN-splittings of the group.
In the fifth chapter, we will take a closer look at the division ring RKG for a free-by-cyclic group G. In this case, according to a result of Linnell and Lück, the ring arises from the group ring KG via Cohn's universal localization. Furthermore, as Jaikin-Zapirain has shown, it is the universal division ring of fractions of the group ring. Using the Farrell--Jones conjecture, which has recently been verified for the class of groups under consideration, we improve upon both of these results by showing that KG is a pseudo-Sylvester domain. This result is also proved more generally for crossed products E∗G with E a division ring of arbitrary characteristic.},

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