A Metric Space Not Quasi-isometrically Embeddable into Any Uniformly Convex Banach Space

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 obstruction to uniform quasi-isometric embeddability (i.e. with embedding constants independent of n) of the l∞’s into any uniformly convex Banach space. From the proof one can see explicitly how the Lipschitz constants of the embedding are being pushed to infinity as n grows larger. In the last section we also comment our observations in view of Bourgain’s paper on superreflexivity [Bo]. The inspiration for the construction of the graph Γ comes from the work of T. Laakso on A∞-weighted metrics on the plane [La], similar spaces were also considered in [NR] in the context of metric embeddings of finite graphs into Hilbert spaces. I would like to greatly thank Piotr Haj lasz from whom I learned about Laakso’s work and my advisor Guoliang Yu for his support and guidance in this research. I am also very grateful to William Johnson for illuminating discussions 2000 Mathematics Subject Classification. Primary 51F99; Secondary 46B20 .