This is an announcement for the paper "Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces" by Assaf Naor, Yuval Peres, Oded Schramm and Scott Sheffield.

Abstract: A metric space $X$ has {\em Markov type/} $2$, if for any reversible finite-state Markov chain ${Z_t}$ (with $Z_0$ chosen according to the stationary distribution) and any map $f$ from the state space to $X$, the distance $D_t$ from $f(Z_0)$ to $f(Z_t)$ satisfies $\E(D_t^2) \le K^2, t, \E(D_1^2)$ for some $K=K(X)<\infty$. This notion is due to K.,Ball (1992), who showed its importance for the Lipschitz extension problem. However until now, only Hilbert space (and its bi-Lipschitz equivalents) were known to have Markov type 2. We show that every Banach space with modulus of smoothness of power type $2$ (in particular, $L_p$ for $p>2$) has Markov type $2$; this proves a conjecture of Ball. We also show that trees, hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature have Markov type $2$. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in 1982, by showing that for $1<q<2<p<\infty$, any Lipschitz mapping from a subset of $L_p$ to $L_q$ has a Lipschitz extension defined on all of $L_p$.

Archive classification: Functional Analysis; Probability

Mathematics Subject Classification: 46B99 (primary), 60B99 (secondary)

Remarks: 27 pages

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Submitted from: anaor@microsoft.com

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