Nonlinear spectral calculus and super-expanders

Abstract

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.Comment: Typos fixed based on referee comments. Some of the results of this paper were announced in arXiv:0910.2041. The corresponding parts of arXiv:0910.2041 are subsumed by the current pape

Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

“…Definition 1.2 is taken from [43]. In [3] Ball suggested a seemingly different notion of Markov cotype, but it is in fact equivalent to Definition 1.2, as explained in Section 7.…”

“…Our proof of Theorem 1.5 is based on an extension of the method of [43] to the present nonlinear setting. In particular we prove for this purpose a nonlinear analogue of Pisier's martingale cotype inequality [57]; see Section 2 below.…”

“…The special case of Theorem 1.5 when X is a Banach space whose modulus of uniform convexity has power type was proved in [3,43]. Our proof of Theorem 1.5 is based on an extension of the method of [43] to the present nonlinear setting.…”

“…The remaining terms in (1.2) correspond to the reversal of (1.1), with { } =1 replaced by { } =1 in the left hand side, and the power A replaced by the Cesàro average 1 =1 A . Due to [3,43], Banach spaces that admit an equivalent norm whose modulus of convexity has power type have metric Markov cotype , in particular N ( ) 1 for ∈ [2 ∞) and N 2 ( ) 1/ − 1 for ∈ (1 2]. Prior to the present work this was the only nontrivial class of metric spaces whose metric Markov cotype was known.…”

“…Due to our computation of metric Markov cotype for barycentric spaces, this yields a versatile Lipschitz extension theorem that contains as special cases many Lipschitz extension theorems that appeared in the literature, as well as Lipschitz extension results that were previously unknown. Another use of metric Markov cotype is due to [43], where it is shown to yield spectral calculus inequalities for nonlinear spectral gaps. Consequently, our computation of metric Markov cotype for barycentric metric spaces implies new nonlinear spectral calculus inequalities which, in the special case of C AT (0) spaces, lay the groundwork for our forthcoming construction [44] of expanders with respect to certain Hadamard spaces and random graphs.…”