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Amenable actions and exactness for discrete groups
 C. R. Acad. Sci. Paris Sér. I Math
"... Abstract. It is proved that a discrete group G is exact if and only if its left translation action on the StoneČech compactification is amenable. Combining this with an unpublished result of Gromov, we have the existence of non exact discrete groups. In [KW], Kirchberg and Wassermann discussed exac ..."
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Cited by 65 (3 self)
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Abstract. It is proved that a discrete group G is exact if and only if its left translation action on the StoneČech compactification is amenable. Combining this with an unpublished result of Gromov, we have the existence of non exact discrete groups. In [KW], Kirchberg and Wassermann discussed exactness for groups. A discrete group G is said to be exact if its reduced group C∗algebra C ∗ λ (G) is exact. Throughout this paper, G always means a discrete group and we identify G with the corresponding convolution operators on ℓ2(G). Amenability of a group action was discussed by AnantharamanDelaroche and Renault in [ADR]. The left translation action of a group G on its StoneČech compactification βG was considered by Higson and Roe in [HR]. This action is amenable if and only if the uniform Roe algebra UC ∗ (G): = C ∗ (ℓ∞(G), G) = span{sℓ∞(G) : s ∈ G} ⊂ B(ℓ2(G)) is nuclear. Since a C ∗subalgebra of an exact C ∗algebra is exact, C ∗ λ (G) is exact if UC ∗ (G) is nuclear. In this article, we will prove the converse. A function u: G×G → C is called a positive definite kernel if the matrix [u(si, sj)] ∈ Mn is positive for any n and s1,..., sn ∈ G. If u is a positive definite kernel on G × G such that u(s, s) ≤ 1 for all
Exactness and uniform embeddability of discrete groups
 J. London Math. Soc
, 2004
"... Abstract. We define a numerical quasiisometry invariant, R(Γ), of a finitely generated group Γ, whose values parametrize the difference between Γ being uniformly embeddable in a Hilbert space and C ∗ r(Γ) being exact. 1. ..."
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Cited by 53 (2 self)
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Abstract. We define a numerical quasiisometry invariant, R(Γ), of a finitely generated group Γ, whose values parametrize the difference between Γ being uniformly embeddable in a Hilbert space and C ∗ r(Γ) being exact. 1.
The Novikov conjecture for linear groups
 Publ. Math. Inst. Hautes Études Sci
"... Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2, K) admits a proper, affine isometric action on a Hilbert ..."
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Cited by 50 (3 self)
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Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2, K) admits a proper, affine isometric action on a Hilbert space. This implies that the BaumConnes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C∗algebra theory.
Metric cotype
, 2005
"... We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either a ..."
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Cited by 44 (22 self)
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We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either almost universal (i.e., contains any finite metric space with any distortion> 1), or there exists α> 0, and arbitrarily large npoint metrics whose distortion when embedded in any member of F is at least Ω ((log n) α). The same property is also used to prove strong nonembeddability theorems of Lq into Lp, when q> max{2, p}. Finally we use metric cotype to obtain a new type of isoperimetric inequality on the discrete torus. 1
Asymptotic topology
, 1999
"... Abstract. We establish some basic theorems in dimension theory and absolute extensor ..."
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Cited by 38 (13 self)
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Abstract. We establish some basic theorems in dimension theory and absolute extensor
Metrics on diagram groups and uniform embeddings in a Hilbert space
 Comment. Math. Helv
"... Abstract. We give first examples of finitely generated groups having an intermediate, with values in (0, 1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups are certain diagram groups. In particular ..."
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Cited by 32 (4 self)
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Abstract. We give first examples of finitely generated groups having an intermediate, with values in (0, 1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups are certain diagram groups. In particular, we show that the Hilbert space compression of Richard Thompson’s group F is equal to 1/2, the Hilbert space compression of Z ≀Z is between 1/2 and 3/4, and the Hilbert space compression of Z ≀ (Z ≀ Z) is between 0 and 1/2. In general we find a relationship between the growth of G and the Hilbert space compression of Z ≀ G. 1.
Constructions preserving Hilbert space uniform embeddability of discrete groups
, 2002
"... Abstract. Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences to the Novikov conjecture. Exactness, introduced and studied extensively by KirchbergWassermann, is a functio ..."
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Cited by 32 (8 self)
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Abstract. Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences to the Novikov conjecture. Exactness, introduced and studied extensively by KirchbergWassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that as properties of countable discrete groups uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions. 1.