Schick, Thomas

[["dc.bibliographiccitation.firstpage","1003"],["dc.bibliographiccitation.issue","4"],["dc.bibliographiccitation.journal","Journal of the American Mathematical Society"],["dc.bibliographiccitation.lastpage","1051"],["dc.bibliographiccitation.volume","20"],["dc.contributor.author","Linnell, Peter A."],["dc.contributor.author","Schick, Thomas"],["dc.date.accessioned","2017-09-07T11:47:09Z"],["dc.date.available","2017-09-07T11:47:09Z"],["dc.date.issued","2007"],["dc.description.abstract","The Atiyah conjecture for a discrete group $ G$ states that the $ L^2ehBBetti numbers of a finite CW-complex with fundamental group $ G$ are integers if $ G$ is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of $ G$. Here we establish conditions under which the Atiyah conjecture for a torsion-free group $ G$ implies the Atiyah conjecture for every finite extension of $ G$. The most important requirement is that $ H^ (G,\\mathbb{Z}/p)$ is isomorphic to the cohomology of the $ pehBadic completion of $ G$ for every prime number $ p$. An additional assumption is necessary e.g. that the quotients of the lower central series or of the derived series are torsion-free. We prove that these conditions are fulfilled for a certain class of groups, which contains in particular Artin's pure braid groups (and more generally fundamental groups of fiber-type arrangements), free groups, fundamental groups of orientable compact surfaces, certain knot and link groups, certain primitive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture, provided the group does. As a consequence, if such an extension $ H$ is torsion-free, then the group ring $ \\mathbb{C}H$ contains no non-trivial zero divisors, i.e. $ H$ fulfills the zero-divisor conjecture. In the course of the proof we prove that if these extensions are torsion-free, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin's full braid group, therefore answering question B6 on http://www.grouptheory.info. Our methods also apply to the Baum-Connes conjecture. This is discussed by Thomas Schick in his preprint ``Finite group extensions and the Baum-Connes conjecture'', where for example the Baum-Connes conjecture is proved for the full braid groups."],["dc.identifier.doi","10.1090/S0894-0347-07-00561-9"],["dc.identifier.gro","3146670"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4460"],["dc.language.iso","en"],["dc.notes.intern","mathe"],["dc.notes.status","final"],["dc.notes.submitter","chake"],["dc.relation.issn","0894-0347"],["dc.title","Finite group extensions and the Atiyah conjecture"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","839"],["dc.bibliographiccitation.issue","7"],["dc.bibliographiccitation.journal","Communications on Pure and Applied Mathematics"],["dc.bibliographiccitation.lastpage","873"],["dc.bibliographiccitation.volume","56"],["dc.contributor.author","Dodziuk, Józef"],["dc.contributor.author","Linnell, Peter A."],["dc.contributor.author","Mathai, Varghese"],["dc.contributor.author","Schick, Thomas"],["dc.contributor.author","Yates, Stuart"],["dc.date.accessioned","2017-09-07T11:47:15Z"],["dc.date.available","2017-09-07T11:47:15Z"],["dc.date.issued","2003"],["dc.description.abstract","Let G be a torsion‐free discrete group, and let ℚ denote the field of algebraic numbers in ℂ. We prove that ℚG fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups that are residually torsion‐free elementary amenable or are residually free. This result implies that there are no nontrivial zero divisors in ℂG. The statement relies on new approximation results for L2‐Betti numbers over ℚG, which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number‐theoretic properties of eigenvalues for the combinatorial Laplacian on L2‐cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the Linnell class 𝒞. We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class 𝒢."],["dc.identifier.doi","10.1002/cpa.10076"],["dc.identifier.gro","3146686"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4477"],["dc.language.iso","en"],["dc.notes.status","final"],["dc.notes.submitter","chake"],["dc.relation.issn","0010-3640"],["dc.title","Approximating ^2ehBinvariants and the Atiyah conjecture"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","261"],["dc.bibliographiccitation.journal","Geometriae Dedicata"],["dc.bibliographiccitation.lastpage","266"],["dc.bibliographiccitation.volume","158"],["dc.contributor.author","Linnell, Peter A."],["dc.contributor.author","Okun, Boris"],["dc.contributor.author","Schick, Thomas"],["dc.date.accessioned","2017-09-07T11:43:05Z"],["dc.date.available","2017-09-07T11:43:05Z"],["dc.date.issued","2012"],["dc.identifier.doi","10.1007/s10711-011-9631-y"],["dc.identifier.gro","3146645"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4432"],["dc.notes.intern","mathe"],["dc.notes.status","public"],["dc.notes.submitter","chake"],["dc.relation.issn","0046-5755"],["dc.title","The strong Atiyah conjecture for right-angled Artin and Coxeter groups"],["dc.type","journal_article"],["dc.type.internalPublication","unknown"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","663"],["dc.bibliographiccitation.issue","9"],["dc.bibliographiccitation.journal","Comptes Rendus de l'Académie des Sciences - Series I - Mathematics"],["dc.bibliographiccitation.lastpage","668"],["dc.bibliographiccitation.volume","331"],["dc.contributor.author","Grigorchuk, Rostislav I."],["dc.contributor.author","Linnell, Peter A."],["dc.contributor.author","Schick, Thomas"],["dc.contributor.author","Żuk, Andrzej"],["dc.date.accessioned","2017-09-07T11:47:15Z"],["dc.date.available","2017-09-07T11:47:15Z"],["dc.date.issued","2000"],["dc.description.abstract","In this Note we explain how the computation of the spectrum of the lamplighter group from [9] yields a counterexample to a strong version of the Atiyah conjectures about the range of L2-Betti numbers of closed manifolds."],["dc.description.abstract","Dans cette Note, on montre comment le calcul du spectre du groupe de l'allumeur de réverbères, fait dans [9], donne un contre-exemple à une des conjectures d'Atiyah sur les nombres de Betti L2 des variétés fermées."],["dc.identifier.doi","10.1016/S0764-4442(00)01702-X"],["dc.identifier.gro","3146698"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4491"],["dc.language.iso","en"],["dc.notes.intern","mathe"],["dc.notes.status","final"],["dc.notes.submitter","chake"],["dc.relation.issn","0764-4442"],["dc.title","On a question of Atiyah"],["dc.type","journal_article"],["dc.type.internalPublication","no"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","315"],["dc.bibliographiccitation.lastpage","321"],["dc.contributor.author","Linnell, Peter A."],["dc.contributor.author","Lück, Wolfgang"],["dc.contributor.author","Schick, Thomas"],["dc.contributor.editor","Farrell, F. T."],["dc.contributor.editor","Lück, Wolfgang"],["dc.date.accessioned","2017-09-07T11:47:14Z"],["dc.date.available","2017-09-07T11:47:14Z"],["dc.date.issued","2003"],["dc.description.abstract","Let G = ℤ/2ℤ ≀ ℤ be the so called lamplighter group and k a commutative ring. We show that kG does not have a classical ring of quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka notebook problem. Assume that kG is contained in a ring R in which the element 1 – x is invertible, with x a generator of ℤ ⊂ G. Then R is not flat over kG. If k = ℂ, this applies in particular to the algebra of unbounded operators affiliated to the group von Neumann algebra of G. We present two proofs of these results. The second one is due to Warren Dicks, who, having seen our argument, found a much simpler and more elementary proof, which at the same time yielded a more general result than we had originally proved. Nevertheless, we present both proofs here, in the hope that the original arguments might be of use in some other context not yet known to us."],["dc.identifier.doi","10.1142/9789812704443_0013"],["dc.identifier.gro","3146684"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4475"],["dc.language.iso","en"],["dc.notes.intern","mathe"],["dc.notes.status","final"],["dc.notes.submitter","chake"],["dc.publisher","World Scientific Publishing"],["dc.relation.conference","ICTP 2001"],["dc.relation.eventend","2001-06-08"],["dc.relation.eventlocation","Trieste"],["dc.relation.eventstart","2001-05-21"],["dc.relation.isbn","978-981-238-223-8"],["dc.relation.ispartof","High-Dimensional Manifold Topology"],["dc.subject","Ore ring affiliated operators flat lamplighter group Fox calculus"],["dc.title","The Ore condition, affiliated operators, and the lamplighter group"],["dc.type","conference_paper"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","3449"],["dc.bibliographiccitation.issue","10"],["dc.bibliographiccitation.journal","Proceedings of the American Mathematical Society"],["dc.bibliographiccitation.lastpage","3459"],["dc.bibliographiccitation.volume","136"],["dc.contributor.author","Blomer, Inga"],["dc.contributor.author","Linnell, Peter A."],["dc.contributor.author","Schick, Thomas"],["dc.date.accessioned","2017-09-07T11:43:11Z"],["dc.date.available","2017-09-07T11:43:11Z"],["dc.date.issued","2008"],["dc.identifier.doi","10.1090/S0002-9939-08-09395-7"],["dc.identifier.gro","3146662"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4450"],["dc.notes.intern","mathe"],["dc.notes.status","public"],["dc.notes.submitter","chake"],["dc.relation.issn","0002-9939"],["dc.title","Galois cohomology of completed link groups"],["dc.type","journal_article"],["dc.type.internalPublication","unknown"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","313"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","Pure and Applied Mathematics Quarterly"],["dc.bibliographiccitation.lastpage","327"],["dc.bibliographiccitation.volume","8"],["dc.contributor.author","Linnell, Peter A."],["dc.contributor.author","Schick, Thomas"],["dc.date.accessioned","2017-09-07T11:43:02Z"],["dc.date.available","2017-09-07T11:43:02Z"],["dc.date.issued","2012"],["dc.description.abstract","Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of C that is closed under complex conjugation. Let U(G) denote the algebra of unbounded operators affiliated to the group von Neumann algebra N(G), and let D(KG,U(G)) denote the division closure of K GinU(G); thus D(KG,U(G)) is the smallest subring of U(G) containing KG that is closed under taking inverses. Suppose n is a positive integer, andα∈Mn(KG). Thenαinduces a bounded linear mapα:`2(G)n→`2(G)n, and ker α has a well-defined von Neumann dimension dimN(G)(kerα). This is a nonnegative real number, and one version of the Atiyah conjecture states that ddimN(G)(kerα)∈Z. Assuming this conjecture, we shall prove that if G has no nontrivial finite normal subgroup, then D(KG,U(G)) is ad×d matrix ring over a skew field. We shall also consider the case when G has a nontrivial finite normal subgroup, and other subrings of U(G) that contain KG."],["dc.identifier.doi","10.4310/PAMQ.2012.v8.n2.a1"],["dc.identifier.gro","3146646"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4434"],["dc.language.iso","en"],["dc.notes.intern","mathe"],["dc.notes.status","final"],["dc.notes.submitter","chake"],["dc.relation.issn","1558-8599"],["dc.title","The Atiyah Conjecture and Artinian Rings"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]