Coarse embeddings and non-positive curvature

Let \((X,d)\) be a complete, geodesic metric space.

  • \((X,d)\) is called an Alexandrov space of global non-positive curvature if for every quadruple of points \(x,y,z,w\) such that \(w\) is a metric midpoint of \(x\) and \(y\), i.e., \(d(w,x) = d(w,y) = d(x,y)/2\), we have \[d(z,w)^2 + d(x,y)^2/4 \le d(z,x)^2/2 + d(z,y)^2/2.\]
  • If the reverse inequality holds true for all such quadruples, then \((X,d)\) is called an Alexandrov space of global non-negative curvature.

We will also need the notion of a coarse embedding. A (not necessarily continuous) map \(f\colon Y \to X\) between metric spaces \((Y,d_Y)\) and \((X,d_X)\) is called a coarse embedding, if there exist two non-decreasing functions \(\omega, \Omega\colon [0,\infty) \to [0,\infty)\) with \(\omega \le \Omega\) point-wise and \(\lim_{t\to\infty}\omega(t) = \infty\), such that \[\forall x,y\in Y \colon \omega(d_Y(x,y)) \le d_X(f(x),f(y)) \le \Omega(d_Y(x,y)).\]

A. Eskenazis, M. Mendel and A. Naor recently proved the following theorem (arXiv:1808.02179): There exists a metric space \(Y\) that does not coarsely embed into any non-positively curved Alexandrov space \(X\).

The interest in such a theorem stems from the following two observations:

  • Admitting a coarse embedding into a Hilbert space (which are Alexandrov spaces of global non-positive curvature) has consequences for versions of the Novikov conjecture. And the known counter-examples to the Baum-Connes conjecture rely on spaces (so-called expanders) which were only considered in this situation, because they do not embed into a Hilbert space. Generalizing then from Hilbert spaces to more general spaces allowed us to discover more phenomena of some problems related to the Novikov and Baum-Connes conjectures, hence studying exotic space which do not admit coarse embeddings into non-positively curved spaces might expose more interesting phenomena.
  • The theorem becomes wrong if we replace the words non-positively curved by non-negatively curved: it follows from arXiv:1509.08677 that every metric space coarsely embeds into some non-negatively curved Alexandrov space (this is explained in Section 1.4.1 of the above cited paper arXiv:1808.02179).

Let me finish by mentioning that the cited paper of Eskenazis-Mendel-Naor has a nice introduction going into many details about the motivation and the history of the above discussed problems, hence is worth reading if you are interested in such things.