Remarks on Quasi-isometric Non-embeddability into Uniformly Convex Banach Spaces

We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of c0 and superreflexivity are discussed. The question of coarse embeddability into uniformly convex Banach spaces became interesting after the recent work of G. Kasparov and G. Yu, who showed the coarse Novikov Conjecture, i.e. that the coarse assembly map in K-theory is injective, for bounded geometry metric spaces coarsely embeddable into uniformly convex Banach spaces [KY]. So far there is no example of a bounded geometry metric space which wouldn’t admit a coarse embedding into any uniformly convex Banach space such spaces are hard to find even if we restrict the target space to be Hilbert. In this note we consider quasi-isometric embeddings, a special case of coarse embedding that may be described as large scale biLipschitz. We construct a locally finite graph which does not admit a quasi-isometric embedding into any uniformly convex Banach space. In contrast to this note that every separable metric space admits a biLipschitz embedding into a strictly convex Banach space, namely into the space c0 with an equivalent strictly convex norm, by a classic result of I. Aharoni [Ah]. It also turns out that our methods can be applied to a c0-type of geometry. As a result we find an explicit geometric obstruction to uniform quasi-isometric embeddability (i.e. with embedding constants independent of n) of the l∞’s into any uniformly convex Banach space. In the proof one can directly see how the Lipschitz constants of the embedding are being pushed away to infinity as n grows larger. In the last section we also comment our observations in view of Bourgain’s paper on superreflexivity [Bo], from which quasi-isometric non-embeddability of trees into uniformly convex Banach spaces can be deduced. Bourgain’s proof is of different nature, using classic characterization of superreflexivity due to R.C. James and G. Pisier. We show that our graph has essentially the opposite of the hyperbolic geometry on the tree and that the two results are independent of each other. 2000 Mathematics Subject Classification. Primary 51F99; Secondary 46B20 .