Markov convexity and local rigidity of distorted metrics (Q1932211)

The first main result of the paper (Theorem 1.3) is a purely metric characterization of the class of uniformly convex Banach spaces whose modulus of convexity is \(\geq ct^p\) for fixed \(p\in[2,\infty)\) and some \(c>0\). The characterization is in terms of the Markov \(p\)-convexity introduced by \textit{J. R. Lee, A. Naor} and \textit{Y. Peres} [Geom. Funct. Anal. 18 (2008), No. 5, 1609--1659 (2009; Zbl 1171.05318)]. (The Markov \(p\)-convexity of a space is defined in terms of the behavior of Markov chains in the space.) The next main result is Theorem 1.4: If \((X,d_X)\) is Markov \(p\)-convex and \((Y,d_Y)\) is a Lipschitz quotient of \(X\), then \(Y\) is also Markov \(p\)-convex. In this result, the authors use the following definition introduced by \textit{M. Gromov} [Metric structures for Riemannian and non-Riemannian spaces. Boston, MA: Birkhäuser (1999; Zbl 0953.53002)]. A surjective map \(f:X\to Y\) between two metric spaces is called a Lipschitz quotient if it is Lipschitz, and is such that there exists a constant \(c>0\) such that, for every \(x\in X\) and \(r>0\), \[ f\left(B_X(x,r)\right)\supseteq B_Y\left(f(x),\frac{r}{c}\right). \] Theorem 1.4 extends some of the results on Lipschitz quotients obtained by \textit{S. Bates} et al. [Geom. Funct. Anal. 9, No. 6, 1092--1127 (1999; Zbl 0954.46014)]. The next result (Theorem 1.5) states that there exists a metric space which is not Markov \(p\)-convex for any \(p\) and yet does not admit bilipschitz embeddings of finite binary trees of all sizes with uniformly bounded distortions. A very important result is Theorem 1.12: Let \(B_\infty\) be the complete unweighted infinite binary tree. For every \(D\geq 4\), there exists a metric \(d\) on \(B_\infty\) that is \(D\)-equivalent to the original shortest-path metric on \(B_\infty\), yet for every \(\varepsilon\in (0,1)\) and \(m\in \mathbb{N}\), \[ c_{(B_\infty,d)}(B_m)\leq D-\varepsilon\implies m\leq D^{CD^2/\varepsilon}. \] In this statement, \(B_m\) denotes the binary tree of depth \(m\) with its usual metric and \(c_{(B_\infty,d)}(B_m)\) denotes the (optimal) distortion of embeddings of \(B_m\) into \((B_\infty,d)\). Theorem 1.12 answers a question asked by C.~Fefferman. This result shows the important difference between the behavior of trees in Banach spaces and in general metric spaces. This paper contains many other important results and an extensive introduction which puts these results in context. The last section contains a short list of open problems. The reader interested in the topic will find in this paper much more than is described in this review.