Nonpositive curvature is not coarsely universal

• 成果类型: Article
• 署名作者: Eskenazis, Alexandros; Mendel, Manor; Naor, Assaf
• 署名单位: Princeton University; Open University Israel
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00878-1
• 发表日期: 2019
• 页码: 833-886
• 关键词: discrete-group-actions; metric-measure-spaces; uniform embeddings; banach-spaces; probabilistic inequalities; poincare inequalities; novikov-conjecture; HYPERBOLIC GROUPS; cat(0) spaces; warped cones

摘要

We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question ofGromov (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 Ec Norm Super (4) 51(3):657-700, 2018). We establish this statement by proving that a metric space which is q- barycentric for some q is an element of [1, infinity) 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 l(infinity) grid [m](infinity)(n) = ({1, ..., m}(n), parallel to center dot parallel to(infinity)) into l(q) for all q is an element of (2, infinity), from which we deduce the following discrete converse to the fact that l(infinity)(n) embeds with distortion O(1) into l(q) for q = O(log n). A rigidity theorem of Ribe (Ark Mat 14(2):237-244, 1976) implies that for every n is an element of N there exists m is an element of N such that if [m](infinity)(n) embeds into l(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.