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

TL;DR: In this paper, it was shown that the strong Atiyah conjecture and the Luck approximation conjecture in the space of marked groups hold for locally indicable groups and that one-relator groups satisfy both conjectures.

Abstract: It is shown that the strong Atiyah conjecture and the Luck 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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01 Jan 2006

TL;DR: In this paper, the authors present a list of special notation for preiminaries on modules over firs and semi-firs, including principal ideal domains, centralizers, subalgebras and Skew fields of fractions.

Abstract: Preface Note to the reader Terminology, notations and conventions used List of special notation 0. Preliminaries on modules 1. Principal ideal domains 2. Firs, semifirs and the weak algorithm 3. Factorization 4. 2-firs with a distributive factor lattice 5. Modules over firs and semifirs 6. Centralizers and subalgebras 7. Skew fields of fractions Appendix Bibliography and author index Subject index.

11 citations

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TL;DR: The existence of Hughes-free division E ∗G-ring DE∗G for an arbitrary locally indicable group G is still an open question as discussed by the authors, even if G is amenable or G is bi-orderable.

Abstract: Let E∗G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E∗G-isomorphism, there exists at most one Hughes-free division E∗G-ring. However, the existence of a Hughes-free division E∗G-ring DE∗G for an arbitrary locally indicable group G is still an open question. Nevertheless, DE∗G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether DE∗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 DE[G] and it is universal. In Appendix we give a description of DE[G] when G is a RFRS group.

5 citations

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TL;DR: In this article , an analogue of the Lück approximation theorem for certain residually finite rationally soluble (RFRS) groups including right-angled Artin groups and Bestvina-Brady groups is presented.

Abstract: Abstract We prove an analogue of the Lück Approximation Theorem in positive characteristic for certain residually finite rationally soluble (RFRS) groups including right-angled Artin groups and Bestvina–Brady groups. Specifically, we prove that the mod p homology growth equals the dimension of the group homology with coefficients in a certain universal division ring and this is independent of the choice of residual chain. For general RFRS groups we obtain an inequality between the invariants. We also consider a number of applications to fibring, amenable category, and minimal volume entropy.

3 citations

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TL;DR: In this article, the authors obtained a whole family of irrational Betti numbers arising from the lamplighter group algebra, where the elements realizing such irrational numbers are described as crossed product algebras.

Abstract: We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $$\ell ^2$$
-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational $$\ell ^2$$
-Betti numbers arising from the lamplighter group algebra $${\mathbb Q}[{\mathbb Z}_2 \wr {\mathbb Z}]$$
. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $$\ell ^2$$
-Betti numbers from the algebras $${\mathbb Q}[{\mathbb Z}_n \wr {\mathbb Z}]$$
, where $$n \ge 2$$
is a natural number. We also apply the techniques developed to the generalized odometer algebra $${\mathcal {O}}({\overline{n}})$$
, where $${\overline{n}}$$
is a supernatural number. We compute its $$*$$
-regular closure, and this allows us to fully characterize the set of $${\mathcal {O}}({\overline{n}})$$
-Betti numbers.

2 citations

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Balliol College

^{1}TL;DR: In this article, the authors studied the problem of unit formation in the ring of rational integers, where the identity of an element in G is always represented by a symbol e. The symbol e, with or without subscripts, will always denote an element.

Abstract: when addition and multiplication are defined in the obvious way, form a ring, the group-ring of G over K, which will be denoted by R (G, K). Henceforward, we suppose that K has the modulus 1, and we denote the identity in G by e0. Then R(G,K) has the modulus l.e0. Since no confusion can arise thereby, the element 1. e in R(G, K) will be written as e, and whenever it is convenient, the elements e0 in G and re0 in R(G, K) as 1 and r respectively. The symbol e, with or without subscripts, will always denote an element in G. If the elements Ex, E2 in R(G, K) satisfy JE1E2 = 1, Ex will be said to be a left unit, and E9 a right unit in R(G, K). If 77 is a left (or right) unit in K, then rje is a left (or right) unit in R(G, K). Such a unit will be described as trivial. The units in R(G, K) form a group if and only if every right unit is also a left unit. This is so, for instance, if both G and K are Abelian. It is also true if G is a finite group, and K is any ring of complex numbers, for then the regular representation| of G can be extended to give an isomorphism of R(G, K) in the ring of ordinary matrices. The first object of this paper is to study units in R(G,C), where C is the ring of rational integers. In § 2 we take G to be a finite Abelian group,

449 citations

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TL;DR: In this paper, it was shown that the p-thL 2 -Betti number of a CW-complex is the limit of the sequence of normal subgroups of finite index whose intersection is trivial.

Abstract: LetX be a finite connectedCW-complex. Suppose that its fundamental group π is residually finite, i.e. there is a nested sequence ... ⊂ Г
m
+ 1 ⊂ Г
m
⊂ ... ⊂ π of in π normal subgroups of finite index whose intersection is trivial. Then we show that thep-thL
2-Betti number ofX is the limit of the sequenceb
p(Xm)/[π:Г
m
] whereb
p(Xm) is the (ordinary)p-th Betti number of the finite covering ofX associated with Г
m
.

261 citations

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28 Jul 1995TL;DR: In this article, the authors present a list of special notations for skew fields, including rational relations, rational identities, rational relations and rational identities of skew fields and rational relations of rational fields of fractions.

Abstract: Preface From the preface to Skew Field Constructions Note to the reader Prologue 1. Rings and their fields of fractions 2. Skew polynomial rings and power series rings 3. Finite skew field extensions and applications 4. Localization 5. Coproducts of fields 6. General skew fields 7. Rational relations and rational identities 8. Equations and singularities 9. Valuations and orderings on skew fields Standard notations List of special notations used throughout the text Bibliography and author index Subject index.

247 citations

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01 Jan 2006TL;DR: In this article, the authors present a list of special notation for preiminaries on modules over firs and semi-firs, including principal ideal domains, centralizers, subalgebras and Skew fields of fractions.

Abstract: Preface Note to the reader Terminology, notations and conventions used List of special notation 0. Preliminaries on modules 1. Principal ideal domains 2. Firs, semifirs and the weak algorithm 3. Factorization 4. 2-firs with a distributive factor lattice 5. Modules over firs and semifirs 6. Centralizers and subalgebras 7. Skew fields of fractions Appendix Bibliography and author index Subject index.

221 citations