The strong Atiyah conjecture for virtually cocompact special groups

Schreve, Kevin

doi:10.1007/s00208-014-1007-9

Cited by 23 publications

(20 citation statements)

References 19 publications

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“…We may now prove that all hyperbolic virtually special groups are good. This was proved by Schreve [17] and later by Minasyan and Zalesskii [13], both using virtual retraction properties; we give another proof using cube complex hierarchies. Hierarchies have also been used to prove goodness in other contexts, in particular, in [5].…”

Section: Theorem 7 ([5 Proposition 36]) An Efficient Amalgamated Fmentioning

confidence: 63%

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Profinite properties of RAAGs and special groups

Kropholler

Wilkes

2016

Bull. London Math. Soc.

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In this paper, we prove that right‐angled Artin groups (RAAGs) are distinguished from each other by their pro‐p completions for any choice of prime p, and that right‐angled Coxeter groups (RACGs) are distinguished from each other by their pro‐2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore, we give an example of a property of the underlying graph of a RAAG that translates to a property of the profinite completion.

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Section: Theorem 7 ([5 Proposition 36]) An Efficient Amalgamated Fmentioning

confidence: 63%

“…Proposition 3 was sufficient to prove the theorem; in fact RACGs are 2-good, so that we have an isomorphism on cohomology in all dimensions. This follows from extension properties of 2-goodness applied to [10,Proposition 9] or from the work on graph products in [17].…”

Section: Introductionmentioning

confidence: 99%

Profinite properties of RAAGs and special groups

Kropholler

Wilkes

2016

Bull. London Math. Soc.

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“…Another application of Agol's theorem, in the context of the Atiyah and Kaplansky zero-divisor conjectures, was provided by [Sch14]. The main result therein, based on the work of Linnell-Schick-Okun and collaborators, see for instance [LOS12], implies the Atiyah conjecture on 2 -Betti numbers for a large class of groups having the Haagerup property, including cubulable hyperbolic groups.…”

Section: Applicationsmentioning

confidence: 98%

A combination theorem for cubulation in small cancellation theory over free products

Martin¹,

Steenbock²

2017

Ann. inst. Fourier

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We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical C (1/6)-small cancellation condition is cubulable. This yields a new large class of relatively hyperbolic groups that can be cubulated, and constitutes the first instance of a cubulability theorem for relatively hyperbolic groups which does not require any geometric assumption on the peripheral subgroups besides their cubulability. We do this by constructing appropriate wallspace structures for such groups, by combining walls of the free factors with walls coming from the universal cover of an associated 2-complex of groups.

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“…Theorem 4.2 (Linnell [Lin]; Schick [Sch1]; Linnell-Okun-Schick [LOS]; Schreve [Sch2]). Let C 1 be the smallest class of groups containing all free groups which is closed under directed unions and extension by elementary amenable groups.…”

Section: Agrarian Groupsmentioning

confidence: 99%

The Bieri–Neumann–Strebel invariants via Newton polytopes

Kielak

2019

Invent. math.

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We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of Sikorav.We offer several applications: we reprove Thurston's theorem on the existence of a polytope controlling the BNS invariants of a 3-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincaré duality groups of type F in dimension 3 and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl.We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the L 2 -torsion polytope of Friedl-Lück is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl-Lück-Tillmann.

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The strong Atiyah conjecture for virtually cocompact special groups

Abstract: We provide new conditions for the Strong Atiyah conjecture to lift to finite group extensions. In particular, we show these conditions hold for cocompact special groups so the Strong Atiyah conjecture holds for virtually cocompact special groups.

Cited by 23 publications

References 19 publications

Profinite properties of RAAGs and special groups

Profinite properties of RAAGs and special groups

A combination theorem for cubulation in small cancellation theory over free products

The Bieri–Neumann–Strebel invariants via Newton polytopes

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