Coarse Universality

Abstract:

The Bourgain index is a tool that can be used to show that if a separable Banach space contains isomorphic copies of all members of a class C then it must contain isomorphic copies of all separable Banach spaces. This can be applied, e.g., to the class C of separable reflexive spaces. Notably, the embedding of each member of C does not witness the universality of X. We investigate a natural coarse analogue of this index which can be used, e.g., to show that a separable metric space that contains coarse copies of all members in certain “small” classes of metric spaces C then X contains a coarse copy of $c_0$ and thus of all separable metric spaces. This is joint work with F. Baudier, G. Lancien, and Th. Schlumprecht.