Geometric and Analytic Properties of Groups

Guentner, Erik; Kaminker, Jerome

doi:10.1007/978-3-540-39702-1_4

Cited by 12 publications

(16 citation statements)

References 11 publications

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“…(ρ + , ρ − )-coarse embedding) of Gn G into Hn H, then passing to a subsequence and reindexing the components, both for Gn G and Hn H, we can assume that there is a (ρ + , ρ − , c)-coarse equivalence (resp. (ρ + , ρ − )-coarse embedding) f n := f | G/Gn from G/G n to H/H n for all n. Moreover, we can assume that f (1 G/Gn ) = 1 H/Hn for all n. We should remark that for finitely generated groups coarse equivalence coincides with quasi-isometry [19].…”

Section: 3mentioning

confidence: 99%

From the geometry of box spaces to the geometry and measured couplings of groups

Das

2018

J. Topol. Anal.

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In this paper, we prove that if two 'box spaces' of two residually finite groups are coarsely equivalent, then the two groups are 'uniform measured equivalent' (UME). More generally, we prove that if there is a coarse embedding of one box space into another box space, then there exists a 'uniform measured equivalent embedding' (UME-embedding) of the first group into the second one. This is a reinforcement of the easier fact that a coarse equivalence (resp. a coarse embedding) between the box spaces gives rise to a coarse equivalence (resp. a coarse embedding) between the groups.We deduce new invariants that distinguish box spaces up to coarse embedding and coarse equivalence. In particular, we obtain that the expanders coming from SL n (Z) can not be coarsely embedded inside the expanders of SL m (Z), where n > m and n, m ≥ 3. Moreover, we obtain a countable class of residually groups which are mutually coarse-equivalent but any of their box spaces are not coarseequivalent.

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Section: 3mentioning

confidence: 99%

From the geometry of box spaces to the geometry and measured couplings of groups

Das

2018

J. Topol. Anal.

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“…In a recent work [1], Arzhantseva and Tessera answer in the negative the following well-known question [8,16]: does coarse embeddability into Hilbert space is preserved under group extensions of finitely generated groups? Their constructions also provide the first example of a finitely generated group which does not coarsely embed into Hilbert space yet does not contain any weakly embedded expander, answering in the affirmative another open problem [1,24].…”

Section: Introductionmentioning

confidence: 99%

The equivariant coarse Baum-Connes conjecture for metric spaces with proper group actions

Deng¹,

Fu²,

Wang³

2021

Preprint

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The equivariant coarse Baum-Connes conjecture interpolates between the Baum-Connes conjecture for a discrete group and the coarse Baum-Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group Γ acts properly and isometrically on a discrete metric space X with bounded geometry, not necessarily cocompact. We show that if the quotient space X/Γ admits a coarse embedding into Hilbert space and Γ is amenable, and that the Γ-orbits in X are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum-Connes conjecture holds for (X, Γ). Along the way, we prove a K-theoretic amenability statement for the Γ-space X under the same assumptions as above, namely, the canonical quotient map from the maximal equivariant Roe algebra of X to the reduced equivariant Roe algebra of X induces an isomorphism on K-theory.

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“…The group Z/2Z≀ G H does not yet answer the above well-known question [DG03,GK04], as the kernel of its surjection onto H is an infinite rank abelian torsion group which is not finitely generated. However, we modify this example to produce a required extension of two finitely generated groups.…”

Section: Introductionmentioning

confidence: 99%

Admitting a Coarse Embedding is Not Preserved Under Group Extensions

Arzhantseva

Tessera

2018

International Mathematics Research Notices

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We construct a finitely generated group which is an extension of two finitely generated groups coarsely embeddable into Hilbert space but which itself does not coarsely embed into Hilbert space. Our construction also provides a new infinite monster group: the first example of a finitely generated group that does not coarsely embed into Hilbert space and yet does not contain a weakly embedded expander.

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Geometric and Analytic Properties of Groups

Cited by 12 publications

References 11 publications

From the geometry of box spaces to the geometry and measured couplings of groups

From the geometry of box spaces to the geometry and measured couplings of groups

The equivariant coarse Baum-Connes conjecture for metric spaces with proper group actions

Admitting a Coarse Embedding is Not Preserved Under Group Extensions

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