Expansion properties of metric spaces not admitting a coarse embedding into a Hilbert space

The main purpose of the paper is to find some expansion properties of locally finite metric spaces which do not embed coarsely into a Hilbert space. The obtained result is used to show that infinite locally finite graphs excluding a minor embed coarsely into a Hilbert space. In an appendix a direct proof of the latter result is given. 2000 Mathematics Subject Classification: Primary: 46B20; Secondary: 05C12, 54E35 A metric space (M, dM) is called locally finite if all balls in it have finitely many elements. We say that (M, dM) has bounded geometry if for each r > 0 there is U(r) < ∞ such that each ball of radius r in M has at most U(r) elements. Let A and B be metric spaces. A mapping f : A → B is called a coarse embedding if there exist non-decreasing functions ρ1, ρ2 : [0,∞) → [0,∞) such that (1) ∀x, y ∈ A ρ1(dA(x, y)) ≤ dB(f(x), f(y)) ≤ ρ2(dA(x, y)); (2) limr→∞ ρ1(r) = ∞. We are interested in conditions under which a locally finite metric space M embeds coarsely into a Hilbert space. See [Gro93], [Roe03], and [Yu06] for motivation and background for this problem. Since, as it is well-known (see e. g. [Ost09, Section 4]), coarse embeddability into a Hilbert space is equivalent to coarse embeddability into L1, we consider coarse embeddability into L1. Locally finite metric space which are not coarsely embeddable into L1 were characterized in [Ost09] and [Tes09]. We reproduce the characterization as it is stated in [Ost09]. Theorem 1 ([Ost09, Theorem 2.4]) Let (M, dM) be a locally finite metric space which is not coarsely embeddable into L1. Then there exists a constant D, depending on M only, such that for each n ∈ N there exists a finite set Mn ⊂ M and a probability measure μn on Mn ×Mn such that • dM(u, v) ≥ n for each (u, v) ∈ suppμn.