Characterization of the Haagerup property by fibred coarse embedding into Hilbert space

Chen, Xiaoman; Wang, Qin; Wang, Xianjin

doi:10.1112/blms/bdt045

Cited by 36 publications

(57 citation statements)

References 14 publications

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“…where m denotes a coarse disjoint union (see [MS13,Definition 2.17.(2)] and Subsection 3.2). Chen-Wang-Wang [CWW13] showed that G admits a fibred coarse embedding into a Hilbert space if and only if G is a-T-menable. They also showed that for a metric space M , if G is a-M -menable, then G admits a fibred coarse embedding into M .…”

Section: Introductionmentioning

confidence: 99%

Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings

Mimura

Sako

2019

Analysis and Geometry in Metric Spaces

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The objective of this series is to study metric geometric properties of disjoint unions of Cayley graphs of amenable groups by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union admits a fibred coarse embedding into a Hilbert space (as a disjoint union) if and only if the Cayley boundary of the sequence in the space of marked groups is uniformly a-T-menable. We furthermore extend this result to ones with other target spaces. By combining our main results with constructions of Osajda and Arzhantseva-Osajda, we construct two systems of markings of a certain sequence of finite groups with two opposite extreme behaviors of the resulting two disjoint unions: With respect to one marking, the space has property A. On the other hand, with respect to the other, the space does not admit fibred coarse embeddings into Banach spaces with non-trivial type (for instance, uniformly convex Banach spaces) or Hadamard manifolds; the Cayley limit group is, furthermore, non-exact.

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Section: Introductionmentioning

confidence: 99%

Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings

Mimura

Sako

2019

Analysis and Geometry in Metric Spaces

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“…Moreover, in [12] G. Kasparov and G. Yu showed that groups admitting coarse embeddings into uniformly convex Banach spaces satisfy the Novikov conjecture. Inspired by the above works, we extend further the result in [6]: for a residually amenable group G, we give a sufficient condition of the proper isometric affine action of G on some uniformly convex Banach space by fibred cofinitelycoarse embeddability of its box families. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 84%

Fibred cofinitely-coarse embeddability of box families and proper isometric affine actions on uniformly convex Banach spaces

Wang

2018

Isr. J. Math.

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“…While it is theoretically appealing that we can, in particular, characterise many geometric properties of groups by geometric properties of their box spaces (so far, one could do that for ‘group‐theoretic’ or ‘equivariant’ properties like amenability or the Haagerup property and, more generally, property

P B

), we are mostly interested in applications of this new notion. These are provided, in the case of asymptotic dimension, by a recent result of Yamauchi , which we formulate at the end of this section using the general language developed here.…”

Section: Asymptotically Faithful Coverings and Piecewise Propertiesmentioning

confidence: 99%

“…but as there is no dependence on y, it simply means that δ(γ) := δ(γ, y) is a homomorphism. In particular, inequality (8) means that the kernel F of δ is finite, as contained in the ball B(e, CA). Similarly, the existence of elements δ 0 ∈ Δ satisfying formula (7) and with uniformly bounded length (for example, |δ 0 | A as above) is equivalent to saying that the subgroup δ(Γ) is of finite index in Δ.…”

Section: Quasi-isometry Induced By One Mapmentioning

confidence: 99%

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Warped cones, (non‐)rigidity, and piecewise properties

Sawicki

2018

Proc. London Math. Soc.

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We prove that if a quasi‐isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi‐isometry of the respective warped cones. For a general quasi‐isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi‐isometric after taking Cartesian products with suitable powers of the integers. Second, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an application, we calculate the asymptotic dimension of a warped cone, improve bounds by Szabó, Wu, and Zacharias and by Bartels on the amenability dimension of actions of virtually nilpotent groups, and give a partial answer to a question of Willett about dynamic asymptotic dimension. In the Appendix, we justify optimality of the aforementioned result on general quasi‐isometries by showing that quasi‐isometric warped cones need not come from quasi‐isometric groups, contrary to the case of box spaces.

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Characterization of the Haagerup property by fibred coarse embedding into Hilbert space

Cited by 36 publications

References 14 publications

Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings

Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings

Fibred cofinitely-coarse embeddability of box families and proper isometric affine actions on uniformly convex Banach spaces

Warped cones, (non‐)rigidity, and piecewise properties

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