The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space

Yu, Guoliang

doi:10.1007/s002229900032

Cited by 482 publications

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“…For every n ∈ N, one can ask what is the largest Hilbertian bi-Lipschitz distortion of an n-point subset of G, and denoting this number by D G ðnÞ, investigate the rate at which D G ðnÞ tends to infinity. This rate does not depend on a specific finite generating set, so it is a genuine algebraic invariant of G. It turns out to be quite difficult to evaluate the asymptotic behavior of D G ðnÞ, and this has been computed in only a few cases, e.g., D G ðnÞ ≍ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi log log n p if G is a free group on at least two generators (5), and D G ðnÞ ≍ ffiffiffiffiffiffiffiffiffi ffi log n p if G is the Heisenberg group (8). Similar questions with Hilbert space replaced by L 1 are of fundamental importance to theoretical computer science (9).…”

Section: What Is Quantitative Geometry?mentioning

confidence: 99%

Quantitative geometry

Naor

2013

Proc. Natl. Acad. Sci. U.S.A.

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Section: What Is Quantitative Geometry?mentioning

confidence: 99%

Quantitative geometry

Naor

2013

Proc. Natl. Acad. Sci. U.S.A.

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“…Property A was defined by Yu [22] as a geometric condition implying the Coarse Baum-Connes Conjecture for discrete metric spaces. Soon after it was introduced it turned out that Property A for finitely generated group G is equivalent to existence of a topologically amenable action on a compact space [11] as well as to exactness of the reduced group C -algebra C r .G/ [9], [18].…”

Section: -Exact Quotientsmentioning

confidence: 99%

“…Determining how large the class of groups possessing Property A is an active area of research due to applications to various problems in geometry and topology, see for example [19], [20], [21], [22], [23]. One of the most natural topics in this context are permanence properties of exact groups.…”

Section: -Exact Quotientsmentioning

confidence: 99%

Zero-in-the-spectrum conjecture on regular covers of compact manifolds

Nowak¹

2009

Comment. Math. Helv.

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Abstract. We prove the zero-in-the-spectrum conjecture for large, regular covers associated to amenable subgroups of fundamental group of a closed manifold N , provided that 1 .N / is C -exact. Mathematics Subject Classification (2000). Primary 58J22; Secondary 58J50.Keywords. Property A, exact group, zero-in-the-spectrum conjecture.The zero-in-the-spectrum conjecture was first formulated by Gromov [6], [7] and asks if the spectrum Laplace-Beltrami operator acting on the square-integrable p-forms on the universal cover of a closed aspherical manifold contains zero. This fact is implied by the Strong Novikov Conjecture and thus the interest in finding a counterexample. A more general zero-in-the-spectrum conjecture on open complete manifolds was stated by Lott and it is true if there is a positive answer to the following question: does the spectrum of the Laplace-Beltrami operator p acting on square-integrable p-forms of a complete manifold M contain zero for some p D 0; 1; : : : ? The answer is negative in general: Farber and Weinberger [4] showed that for every n 6 there exists a manifold N such that zero is not in the spectrum of p for any p 2 0; 1; : : : acting on the universal cover of N . Later Higson, Roe and Schick [12] extended this result and gave a complete description of groups which can appear as fundamental groups of manifolds whose universal covers do not have zero in the spectrum of the Laplacian.Because of the origins of the problem, various covering spaces are a natural environment for considering zero-in-the-spectrum questions. An early result of this type is a theorem of Brooks [1] stating that given a regular cover M of a compact manifold N , 0 is in the spectrum of 0 on M if and only if the group of deck transformations is amenable. The articles [14], [15] provide a comprehensive survey of this topic.The purpose of this article is to prove the zero-in-the-spectrum conjecture on certain regular covers associated to normal, amenable subgroups of the fundamental

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“…Exact groups are known to admit uniform embeddings in a Hilbert space and therefore to satisfy the Novikov conjecture by an important result of Yu [26]. Here we have a concrete application of noncommutative geometry to a classical problem in topology.…”

Section: ])mentioning

confidence: 96%

“…In particular, since the CPAP implies the CBAP, Lance's theorem implies that amenable groups satisfy the Novikov conjecture. On the other hand there exist exact groups that are not amenable, for example the free groups [13] and the word hyperbolic groups [26].…”

Section: Theorem 39 If a C * -Algebra A Satisfies The Cbap Then A Imentioning

confidence: 99%

Approximation Properties for Discrete Groups

Brodzki

Niblo

Trends in Mathematics

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The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space

Cited by 482 publications

References 23 publications

Quantitative geometry

Quantitative geometry

Zero-in-the-spectrum conjecture on regular covers of compact manifolds

Approximation Properties for Discrete Groups

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