Vanishing of $\ell ^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence

Sauer, Roman; Schrödl, Michael

doi:10.4064/fm512-1-2018

Cited by 11 publications

(20 citation statements)

References 23 publications

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“…All these classes are closed under passing to quotients by finite normal subgroups: For (1') the closure property follows from the fact that the vanishing of ℓ 2 -Betti numbers is an invariant of virtual isomorphism which can be easily deduced from their basic properties [43]. Or one may cite the much more general result that the vanishing of ℓ 2 -Betti numbers of lcsc groups is a coarse invariant [60].…”

Section: Conclusion Of Proof Of Theorem 14mentioning

confidence: 99%

Lattice envelopes

2020

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We introduce a class of countable groups by some abstract grouptheoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing ℓ 2 -Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group Γ in this class we determine the general structure of the possible lattice embeddings of Γ, i.e. of all compactly generated, locally compact groups that contain Γ as a lattice. This leads to a precise description of possible non-uniform lattice embeddings of groups in this class. Further applications include the determination of possible lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature.

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Section: Conclusion Of Proof Of Theorem 14mentioning

confidence: 99%

Lattice envelopes

2020

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“…Topological orbit equivalence as in Theorem and thus bi‐Lipschitz equivalence of finitely generated groups is a strong condition that implies quasi‐isometry. This is known to imply, for example, that cohomological dimensions agree provided they are finite and that (under suitable finiteness conditions) for all

k \in N

the

k

ℓ^{2}

‐Betti number of one groups vanishes if and only if it vanishes for the other due to an unpublished result of Pansu (see also ). In the next section, we will discuss how symmetries of the orbit equivalence relation yield symmetries of certain cohomological invariants of the acting groups.…”

Section: Gromov's Constructionmentioning

confidence: 99%

Cantor systems and quasi-isometry of groups

Medynets

Sauer

Thom

2017

Bull. London Math. Soc.

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The purpose of this note is twofold. In the first part we observe that two finitely generated non-amenable groups are quasi-isometric if and only if they admit topologically orbit equivalent Cantor minimal actions. In particular, free groups if different rank admit topologically orbit equivalent Cantor minimal actions unlike in the measurable setting. In the second part we introduce the measured orbit equivalence category of a Cantor minimal system and construct (in certain cases) a representation of this category on the category of finite-dimensional vector spaces. This gives rise to novel fundamental invariants of the orbit equivalence relation together with an ergodic invariant probability measure.Comment: incorporated referee's suggestions and correction

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“…The equivalence between the group L p -cohomology and the asymptotic L p -cohomology was established by G. Elek [Ele98] for finitely generated groups, and by R. Tessera [Tes09] in degree 1. We recently learnt from R. Sauer and M. Schrödl that they established the coarse invariance of vanishing of 2 -Betti numbers for unimodular locally compact second countable groups [SS18]. To prove this result they established a coarse equivalence version of Theorem 1.1, by using the same comparison strategy -see [SS18,Theorem 12].…”

Section: Introductionmentioning

confidence: 99%

“…We recently learnt from R. Sauer and M. Schrödl that they established the coarse invariance of vanishing of 2 -Betti numbers for unimodular locally compact second countable groups [SS18]. To prove this result they established a coarse equivalence version of Theorem 1.1, by using the same comparison strategy -see [SS18,Theorem 12].…”

Section: Introductionmentioning

confidence: 99%

Quasi-isometric invariance of continuous group L p -cohomology, and first applications to vanishings

Bourdon¹,

Rémy²

2020

Annales Henri Lebesgue

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(*) M.B. was partially supported by the Labex Cempi and B.R. by the GeoLie project (ANR-15-CE40-0012). Both authors were supported by the GDSous/GSG project (ANR-12-BS01-0003). Marc BOURDON & Bertrand RÉMY Résumé.-Nous prouvons que la cohomologie L p continue des groupes localement compacts à base dénombrable d'ouverts est un invariant de quasi-isométrie. En guise d'applications, nous obtenons des résultats partiels dans le sens d'une réponse positive à une question posée par M. Gromov, suggérant un comportement classique de la cohomologie L p continue des groupes de Lie réels simples. Outre l'invariance par quasi-isométrie, les ingrédients utilisés sont un argument de suite spectrale et des résultats d'annulation pour les espaces hyperboliques dus à P. Pansu. Dans les cas de groupes de Lie les mieux adaptés, nous obtenons la moitié des annulations attendues.

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Vanishing of $\ell ^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence

Cited by 11 publications

References 23 publications

Lattice envelopes

Lattice envelopes

Cantor systems and quasi-isometry of groups

Quasi-isometric invariance of continuous group L p -cohomology, and first applications to vanishings

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