On embedding left-ordered groups into division rings

Gräter, Joachim; Sperner, Robert P.

doi:10.1515/forum-2012-0070

Cited by 3 publications

(3 citation statements)

References 23 publications

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“…In this respect it is worth mentioning that unlike us [C] and [KM] deal with right-orders (cf. [GS,Section 2]).…”

Section: Conradian Groupsmentioning

confidence: 99%

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Free division rings of fractions of crossed products of groups with Conradian left-orders

Gräter

2020

Forum Mathematicum

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Let D be a division ring of fractions of a crossed product F [G, η, α] where F is a skew field and G is a group with Conradian left-order ≤. For D we introduce the notion of freeness with respect to ≤ and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F -vector space F ((G)) of all formal power series in G over F with respect to ≤. From this we obtain that all division rings of fractions of F [G, η, α] which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, F [G, η, α] possesses a division ring of fraction which is free in this sense if and only if the rational closure of F [G, η, α] in the endomorphism ring of the corresponding right F -vector space F ((G)) is a skew field.

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“…In this respect it is worth mentioning that unlike us [C] and [KM] deal with right-orders (cf. [GS,Section 2]).…”

Section: Conradian Groupsmentioning

confidence: 99%

“…, h n C ′ are pairwise different. We finally apply our results to a locally indicable group G with a left-order which is not Conradian and refer the reader to [D2,D3,GS] for further details. Let G be the knot group of the trefoil which is isomorphic to the braid group B 3 on 3 strands.…”

Section: Hughes' Theorems and An Examplementioning

confidence: 99%

Free division rings of fractions of crossed products of groups with Conradian left-orders

Gräter

2020

Forum Mathematicum

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“…Thus, a group algebra of a locally indicable group over a field of characteristic zero is embedded in a division algebra. This solves the Malcev problem for this class of group algebras (for more details about this problem see [13]).…”

mentioning

confidence: 94%

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

Jaikin–Zapirain

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.

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On central extensions of SL(2,F) admitting left-orderings

Brungs¹,

Gräter²

2017

Journal of Algebra

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On embedding left-ordered groups into division rings

Cited by 3 publications

References 23 publications

Free division rings of fractions of crossed products of groups with Conradian left-orders

Free division rings of fractions of crossed products of groups with Conradian left-orders

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

On central extensions of SL(2,F) admitting left-orderings

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