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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.
Embeddings of discrete groups and the speed of random walks
, 2007
"... Let G be a group generated by a finite set S and equipped with the associated leftinvariant word metric dG. For a Banach space X let α ∗ X (G) (respectively α # X (G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f: G → X and c> ..."
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Cited by 26 (5 self)
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Let G be a group generated by a finite set S and equipped with the associated leftinvariant word metric dG. For a Banach space X let α ∗ X (G) (respectively α # X (G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f: G → X and c> 0 such that for all x, y ∈ G we have ‖ f (x) − f (y) ‖ ≥ c · dG(x, y) α. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is α ∗ (G) ≔ α ∗ (G) (respectively
Compression functions of uniform embeddings of groups into Hilbert and Banach spaces
 J. Reine Angew. Math
"... We construct finitely generated groups with arbitrary prescribed Hilbert space compression α ∈ [0, 1]. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the Ecompression of these groups coincides with their Hilbert space compression. Moreover, the groups that we c ..."
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Cited by 25 (1 self)
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We construct finitely generated groups with arbitrary prescribed Hilbert space compression α ∈ [0, 1]. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the Ecompression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 3, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (respectively, exact, of finite asymptotic dimension) with Hilbert space compression 0 are given. 1
Property A and CAT(0) cube complexes
 Journal of Functional Analysis
"... Abstract. Property A is a nonequivariant analogue of amenability defined for metric spaces. Euclidean spaces and trees are examples of spaces with Property A. Simultaneously generalizing these facts, we show that finite dimensional CAT(0) cube complexes have Property A. We do not assume that the c ..."
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Cited by 13 (2 self)
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Abstract. Property A is a nonequivariant analogue of amenability defined for metric spaces. Euclidean spaces and trees are examples of spaces with Property A. Simultaneously generalizing these facts, we show that finite dimensional CAT(0) cube complexes have Property A. We do not assume that the complex is locally finite. We also prove that given a discrete group acting properly on a finite dimensional CAT(0) cube complex the stabilisers of vertices at infinity are amenable.
Compression of uniform embeddings into Hilbert space. Arxiv GR/0509108
, 2005
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Lp compression, traveling salesmen, and stable walks
, 2009
"... We show that if H is a group of polynomial growth whose growth rate is at least quadratic then the Lp compression of the wreath product Z ≀ H equals max {} ..."
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Cited by 7 (3 self)
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We show that if H is a group of polynomial growth whose growth rate is at least quadratic then the Lp compression of the wreath product Z ≀ H equals max {}
Some notes on Property A
, 2008
"... The coarse BaumConnes conjecture states that a certain coarse assembly map ..."
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Cited by 6 (2 self)
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The coarse BaumConnes conjecture states that a certain coarse assembly map
The Poisson boundary of CAT(0) cube complex groups
"... Abstract. We consider a finitedimensional, locally finite CAT(0) cube complex X admitting a cocompact properly discontinuous countable group of automorphisms G. We construct a natural compact metric space B(X) on which G acts by homeomorphisms, the action being minimal and strongly proximal. Furt ..."
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Cited by 5 (0 self)
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Abstract. We consider a finitedimensional, locally finite CAT(0) cube complex X admitting a cocompact properly discontinuous countable group of automorphisms G. We construct a natural compact metric space B(X) on which G acts by homeomorphisms, the action being minimal and strongly proximal. Furthermore, for any generating probability measure on G, B(X) admits a unique stationary measure, and when the measure has finite logarithmic moment, it constitutes a compact metric model of the Poisson boundary. We identify a dense Gδ subset UNT (X) of B(X) on which the action of G is Borelamenable, and describe the relation of these two spaces to the Roller boundary. Our construction can be used to give a simple geometric proof of Property A for the complex. Our methods are based on direct geometric arguments regarding the asymptotic behavior of halfspaces and their limiting ultrafilters, which are of considerable independent interest. In particular we analyze the notions of median and interval in the complex, and use the latter in the proof that B(X) is the Poisson boundary via the strip criterion developed by V. Kaimanovich [K]. 1.
Some Remarks on Generalized Roundness
 Geom. Dedicata
"... Abstract. By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank ≥ 2 equals 1. This answers a questio ..."
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Abstract. By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank ≥ 2 equals 1. This answers a question of JF. Lafont and S. Prassidis.
EXACTNESS OF FREE AND AMENABLE GROUPS BY THE CONSTRUCTION OF OZAWA KERNELS
, 2005
"... Abstract. Using properties of their Cayley graphs, specific examples of Ozawa kernels are constructed for both free and amenable groups, thus showing that these groups satisfy Property O. It is deduced both that these groups are exact and satisfy Yu’s Property A. 1. ..."
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Abstract. Using properties of their Cayley graphs, specific examples of Ozawa kernels are constructed for both free and amenable groups, thus showing that these groups satisfy Property O. It is deduced both that these groups are exact and satisfy Yu’s Property A. 1.