L2-Betti Numbers and their Analogues in Positive Characteristic

Advances in Mathematics

Shusterman

2019

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Section: 2mentioning

confidence: 99%

The Hanna Neumann conjecture for Demushkin groups

Advances in Mathematics

Shusterman

2019

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“…In this section we recall the notions of * -regular ring and Sylvester rank function, and explain the main results about epic * -regular R-rings. More information about these topics can be found in [18,19] By a * -regular ring U we mean a von Neumann regular ring together with a proper involution (i.e. an involution * : U → U for which x * x = 0 implies x = 0).…”

Section: * -Regular Sylvester Rank Functionsmentioning

confidence: 99%

“…Sylvester rank functions. The notions of Sylvester matrix rank function rk and Sylvester module rank function (on finitely presented modules) dim were introduced in [28], and to learn more about its properties in our setting one can consult [19,Section 5].…”

Section: 3mentioning

confidence: 99%

“…The notion of natural extension was introduced in [18] in the context of (Laurent) polynomial rings (see also [19,Section 8] for other variations of this concept). In this section we define the natural extension in the context of skew (Laurent) polynomial rings and we use it to define the notion of Hughes-free Sylvester rank function.…”

Section: Natural Extensions and Hughes-free Sylvester Rank Functionsmentioning

confidence: 99%

“…During the last 25 years it has been shown that many families of groups satisfy the strong Atiyah conjecture. We refer the reader to a recent survey [19] of the first author, where all these results are described. In this paper we show that the strong Atiyah conjecture over C holds for locally indicable groups.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The strong Atiyah and Lück approximation conjectures for one-relator groups

López-Álvarez

2019

Math. Ann.

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It is shown that the strong Atiyah conjecture and the Lück approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups.As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.

The universality of Hughes-free division rings

2021

Sel. Math. New Ser.

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Let $$E*G$$ E ∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to $$E*G$$ E ∗ G -isomorphism, there exists at most one Hughes-free division $$E*G$$ E ∗ G -ring. However, the existence of a Hughes-free division $$E*G$$ E ∗ G -ring $${\mathcal {D}}_{E*G}$$ D E ∗ G for an arbitrary locally indicable group G is still an open question. Nevertheless, $${\mathcal {D}}_{E*G}$$ D E ∗ G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether $${\mathcal {D}}_{E*G}$$ D E ∗ G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists $${\mathcal {D}}_{E[G]}$$ D E [ G ] and it is universal. In Appendix we give a description of $${\mathcal {D}}_{E[G]}$$ D E [ G ] when G is a RFRS group.