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Coarse Embeddings Into a Hilbert Space, Haagerup Property and Poincar´Einequalities

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Coarse Embeddings Into a Hilbert Space, Haagerup Property and Poincar´Einequalities Coarse embeddings into a Hilbert space, Haagerup Property and Poincar´einequalities Romain Tessera∗ December 5, 2008 Abstract We prove that a metric space does not coarsely embed into a Hilbert space if and only if it satisfies a sequence of Poincar´einequalities, which can be formulated in terms of (generalized) expanders. We also give quan- titative statements, relative to the compression. In the equivariant context, our result says that a group does not have the Haagerup property if and only if it has relative property T with respect to a family of probabili- ties whose supports go to infinity. We give versions of this result both in terms of unitary representations, and in terms of affine isometric actions on Hilbert spaces. 1 Introduction 1.1 Obstruction to coarse embeddings The notion of expanders has been pointed out by Gromov as an obstruction for a metric space to coarsely embed into a Hilbert space. Recall [JS] (see also [L]) that a sequence of expanders is a sequence of finite connected graphs (Xn) with 2 bounded degree, satisfying the following Poincar´einequality for all f 2 ` (Xn) 1 X 2 C X 2 2 jf(x) − f(y)j ≤ jf(x) − f(y)j ; (1.1) jXnj jXnj x;y2Xn x∼y for some constant C > 0, and whose cardinality jXnj goes to infinity when n ! 1. An equivalent formulation in `p [M2] can be used to prove that expanders do not coarsely embed into Lp for any 1 ≤ p < 1: ∗The author is supported by the NSF grant DMS-0706486. 1 It is an open problem whether a metric space with bounded geometry that does not coarsely embed into a Hilbert space admits a coarsely embedded se- quence of expanders. In this paper, we prove that a metric space (not necessarily with bounded geometry) that does not coarsely embed into a Hilbert space admits a coarsely embedded sequence of \generalized expanders". This weaker notion of expanders can be roughly described as a sequence of Poincar´einequalities with respect to finitely supported probability measures on X × X. We also provide similar obstructions for coarse embeddings into families of metric spaces such as Lp, for every 1 ≤ p < 1, uniformly convex Banach spaces, and CAT(0) spaces. For the sake of clarity, we chose to present most of our results first in the case of Hilbert spaces. However, our characterization (see Theorem 14) of the non-existence of coarse embedding into Lp deserves some attention. Indeed, our Poincar´einequalities are not equivalent for different values of 2 ≤ p < 1. This follows from a result of Naor and Mendel [MN] (see also [JR]) saying that Lp does not coarsely embed into Lq if 2 ≤ q < p. This is different from what happens with real expanders, as having a sequence of expanders prevents from having a coarse embedding into Lp, for any 1 ≤ p < 1. In particular, at least without any assumption of bounded geometry, our generalized expanders cannot be replaced by actual expanders. To conclude, let us remark that finding subspaces of Lp for some p > 2, with bounded geometry, which do not coarsely embed into L2, would answer negatively the problem mentioned above. 1.2 Obstruction to Haagerup Property A countable group is said to have the Haagerup property if it admits a proper affine action on a Hilbert space. An obstruction for an infinite countable group to have the Haagerup Property is known as Property T (also called Property FH), which says that every isometric affine action has a fixed point (or equivalently bounded orbits). A weaker obstruction is to have relative property T with respect to an infinite subset [C1, C2]. The case where this subset is a normal subgroup has been mostly considered, as it has strong consequences. On the other hand, there are examples of groups which do not have relative property T with respect to any subgroup, but have it with respect to some infinite subset [C1]. The question whether the latter property is equivalent to the negation of Haagerup Property is still open. In this paper, we partially answer this question by showing that a countable group does not have Haagerup Property if and only if it has relative Property T 2 with respect to a sequence of probabilities whose supports eventually leave every finite subset. Acknowledgments. I am grateful to Yann Ollivier, Yves Stalder, Yves de Cor- nulier, Bogdan Nica and the referee for their useful comments and corrections. I address a special thank to James Lee who pointed to me [M1, Proposition 15.5.2]. 2 Organization of the paper Before describing the general organization of the paper, let us emphasize that the proof of the main result (stated in its most general form in Theorem 14) is only given in the last section (Section 5). The proof of Theorem 20, relative to length functions on groups, and the proofs of the quantitative statements of Section 4.3 are straightforward adaptations of the proof of Theorem 14. Therefore they will be omitted. • In the next section, we will state our results in the Hilbert case. First, in Subsection 3.1, we state our characterization of the non-existence of a coarse embedding into a Hilbert space. In Subsection 3.2, we show that a sequence of expanders in the usual sense is also a sequence of general- ized expanders in our sense. Finally, in Subsection 3.3, we state various equivalent formulations of our characterization of non-Haagerup property. • In Section 4, we state a more general version of our main result in order to characterize the non-(coarse)-embeddability into various classes of metric spaces, such as Lp-spaces. This section is divided into three main subsec- tions. Subsection 4.1 introduces the notion of sheaves of metrics in order to study coarse embeddings into various classes of metric spaces. We then state our main theorem. Subsection 4.2 is just an adaptation of the defi- nitions and statements of Subsection 4.1 for groups, replacing metrics by length functions (or left-invariant metrics). Finally, in Subsection 4.3, we give quantitative statements relative to the compression of coarse embed- dings into different classes of metric spaces. • The last section is dedicated to the proof of our main result, namely The- orem 14. 3 3 Statement of results in the Hilbert case In this section, we state our main results concerning embeddings into a Hilbert space. In Section 4, using a slightly more sophisticated vocabulary, we generalize to other geometries. 3.1 Coarse embeddability into Hilbert spaces and gener- alized expanders Let H denote a separable infinite dimensional Hilbert space. We denote by jvj the norm of a vector in H: Let X = (X; d) be a metric space. For all r ≥ 0, denote 2 ∆r(X) = f(x; y) 2 X ; d(x; y) ≥ rg: In this paper, we prove that a metric space that does not coarsely embed into a Hilbert space contains in a weak sense a sequence of expanders. Precisely, following the idea of [T, Section 4.2], let us define Definition 1. • Let K and r be positive numbers. A finite metric space is called a general- ized (K; r)-expander if there exists a symmetric probability measure µ sup- ported on ∆r(X) with the following property. For every map F : X !H satisfying kF (x) − F (y)k ≤ d(x; y) for all (x; y) 2 ∆1(X), we have X 2 2 Varµ(F ) := kF (x) − F (y)k µ(x; y) ≤ K : (3.1) x;y • A sequence of finite metric spaces (Xn) is called a sequence of generalized K-expander if for every n 2 N, Xn is a (K; rn)-expanders, where rn ! 1: Remark 2. Note that (3.1) says that F sends certain pairs at distance at least rn in Xn to pairs at distance at most K. Recall that a family of metric spaces (Xi)i2I coarsely embeds into a metric space Y if there exists a family (Fi) of uniformly coarse embeddings of Xi into Y , i.e. if there are two increasing, unbounded functions ρ− and ρ+ such that ρ−(d(x; y)) ≤ d(Fi(x);Fi(y)) ≤ ρ+(d(x; y)); 8x; y 2 Xi; 8i 2 I: (3.2) Proposition 3. A sequence of generalized expanders (Xn) does not coarsely em- bed into a Hilbert space. 4 Proof: Let K > 0 and for all n 2 N, let Xn is a (K; rn)-expander, with rn ! 1. For every n 2 N, let Fn be a map from Xn !H, and that there exists an increasing function ρ+ such that jFn(x) − Fn(y)j ≤ ρ+(d(x; y)); 8x; y 2 Xn; 8n 2 N: As observed in [CTV, Lemmas 2.4 and 3.11], if a metric space (or a family of metric spaces) coarsely embeds into a Hilbert space, we can always assume that the function ρ+ goes arbitrarily slowly to infinity (this follows from a result of Bochner and Schoenberg [Sch, Theorem 8]). So in particular, we can assume that ρ+(t) ≤ t; 8t ≥ 1. But then, (3.1) tells us that pairs of points of Xn, which are at distance ≥ rn are sent by Fn at distance less than K. As rn ! 1, this implies that any increasing function ρ− satisfying ρ−(d(x; y)) ≤ jFn(x) − Fn(y)j; 8x; y 2 Xn; 8n 2 N would have to be ≤ K: Our main result is the following theorem (which is a particular case of Corol- lary 17). Theorem 4. A metric space does not coarsely embed into a Hilbert space if and only if it has a coarsely-embedded sequence of generalized expanders.
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