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Nonpositive curvature is not coarsely universal

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Abstract

We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (Geometric group theory, Cambridge University Press, Cambridge, 1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni et al. (Ann Sci Éc Norm Supér (4) 51(3):657–700, 2018). We establish this statement by proving that a metric space which is q-barycentric for some \(q\in [1,\infty )\) has metric cotype q with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the \(\ell _\infty \) grid \([m]_\infty ^n=(\{1,\ldots ,m\}^n,\Vert \cdot \Vert _\infty )\) into \(\ell _q\) for all \(q\in (2,\infty )\), from which we deduce the following discrete converse to the fact that \(\ell _\infty ^n\) embeds with distortion O(1) into \(\ell _q\) for \(q=O(\log n)\). A rigidity theorem of Ribe (Ark Mat 14(2):237–244, 1976) implies that for every \(n\in {\mathbb {N}}\) there exists \(m\in {\mathbb {N}}\) such that if \([m]_\infty ^n\) embeds into \(\ell _q\) with distortion O(1), then q is necessarily at least a universal constant multiple of \(\log n\). Ribe’s theorem does not give an explicit upper bound on this m, but by the work of Bourgain (Geometrical aspects of functional analysis (1985/86), Springer, Berlin, 1987) it suffices to take \(m=n\), and this was the previously best-known estimate for m. We show that the above discretization statement actually holds when m is a universal constant.

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Notes

  1. Strictly speaking, the above definition is of Alexandrov spaces of global nonpositive curvature, also known as \(\text {CAT}(0)\) spaces or Hadamard spaces. See [27] for the local counterpart of this definition; we will not treat it here, and therefore it will be convenient to drop the term “global” throughout the present text, because the ensuing results are vacuously false under the weaker local assumption (since the 1-dimensional simplicial complex that is associated to any connected combinatorial graph is an Alexandrov space of local nonpositive curvature). For nonnegative curvature, the local and global notions coincide due to the (metric version of the) Alexandrov–Toponogov theorem; see e.g. [152].

  2. In addition to the usual “\(O(\cdot ),o(\cdot )\)” asymptotic notation, it will be convenient to use throughout this article the following (also standard) asymptotic notation. Given two quantities \(Q,Q'>0\), the notations \(Q\lesssim Q'\) and \(Q'\gtrsim Q\) mean that \(Q\leqslant CQ'\) for some universal constant \(C>0\). The notation \(Q\asymp Q'\) stands for \((Q\lesssim Q') \wedge (Q'\lesssim Q)\). If we need to allow for dependence on parameters, we indicate this by subscripts. For example, in the presence of auxiliary objects (e.g. numbers or spaces) \(\phi ,\mathfrak {Z}\), the notation \(Q\lesssim _{\phi ,\mathfrak {Z}} Q'\) means that \(Q\leqslant C(\phi ,\mathfrak {Z})Q' \), where \(C(\phi ,\mathfrak {Z}) >0\) is allowed to depend only on \(\phi ,\mathfrak {Z}\); similarly for the notations \(Q\gtrsim _{\phi ,\mathfrak {Z}} Q'\) and \(Q\asymp _{\phi ,\mathfrak {Z}} Q'\).

  3. A shorter proof of this fact, using an ultrapower and differentiation argument, follows form [76]. Because for this particular case of Ribe’s theorem the target space is \(\ell _q\), a further simplification of the differentiation step is possible; see also [19, 111].

  4. Specifically, in [71, §15(b)] Gromov wrote “The geodesic property is one logical level up from concentration inequalities as it involves the existential quantifier. It is unclear if there is a simple \(\exists \)-free description of (nongeodesic!) subspaces in \(\text {CAT}(\kappa )\)-spaces.” The term “concentration inequalities” is defined in [71, §15(a)] to be the same inequalities as the quadratic metricinequalities that we consider in (12), except that in [71] they are allowed to involve arbitrary powers of the pairwise distances. However, due to [4] it suffices to consider only quadratic inequalities for the purpose of the simple intrinsic description that Gromov hopes to obtain (though, as he indicates, it may not exist).

  5. This demonstrates that the role of the scaling parameter in the definition of metric cotype is not as subsidiary as it may seem from [121], where it did not have a crucial role in the metric characterization of Rademacher cotype or the nonlinear Maurey–Pisier theorem. While sharp metric cotype was shown in [121] to have implications to coarse and uniform embeddings, there is a definite possibility that metric cotype q and sharp metric cotype q coincide for Banach spaces (though this is a major open problem); here we see that this is markedly not so in the setting of Alexandrov geometry, leading to formidable qualitative differences between the coarse implications of the sign of curvature.

  6. In terms of bi-Lipschitz distortion of arbitrary expanders, Table 1 does not fully indicate the extent to which nonnegative curvature behaves better than nonpositive curvature. The fact [141] that an Alexandrov space of nonnegative curvature has Markov type 2 implies (by examining the standard random walk on the graph) that if \(\mathcal {G}=\{\mathsf {G}(n)\}_{n=1}^\infty \) is an expander, then \(\mathsf {c}_X(\mathsf {G}(n))\gtrsim _{\mathcal {G}} \sqrt{\log |\mathsf {G}(n)|}\) for any nonnegatively curved Alexandrov space X. In contrast, there is [96] an Alexandrov space of nonpositive curvature which contains an expander with O(1) distortion.

  7. The only additional observation that is needed for this is that, because the degree of \(\mathsf {G}(n)\) is O(1), a quick and standard counting argument (see e.g. the justification of equation (36) in [134]) shows that \(d_{\mathsf {G}(n)}(u,v)\gtrsim \log |C_j(n)|\rightarrow \infty \) for a constant fraction of \((u,v)\in C_j(n)\times C_j(n)\).

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  1. Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Alexandros Eskenazis & Assaf Naor

  2. Mathematics and Computer Science Department, The Open University of Israel, 1 University Road, P.O. Box 808, 43107, Raanana, Israel

    Manor Mendel

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Alexandros Eskenazis was supported by BSF Grant 2010021 and the Simons Foundation. Manor Mendel was supported by BSF Grant 2010021. Assaf Naor was supported by NSF Grant CCF-1412958, BSF Grant 2010021, the Packard Foundation and the Simons Foundation. This work was carried out under the auspices of the Simons Algorithms and Geometry (A&G) Think Tank, and was completed while Assaf Naor was a member of the Institute for Advanced Study.

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Eskenazis, A., Mendel, M. & Naor, A. Nonpositive curvature is not coarsely universal. Invent. math. 217, 833–886 (2019). https://doi.org/10.1007/s00222-019-00878-1

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