The coarse geometry of Tsirelson’s space and applications

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The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson’s original space T ∗ T^* . Every Banach space that is coarsely embeddable into T ∗ T^* must be reflexive, and all of its spreading models must be isomorphic to c 0 c_0 . Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: T ∗ T^* coarsely contains neither c 0 c_0 nor ℓ p \ell _p for p ∈ [ 1 , ∞ ) p\in [1,\infty ) . We show that there is no infinite-dimensional Banach space that coarsely embeds into every infinite-dimensional Banach space. In particular, we disprove the conjecture that the separable infinite-dimensional Hilbert space coarsely embeds into every infinite-dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs that take values into T ∗ T^* , and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to c 0 c_0 . Also, a purely metric characterization of finite dimensionality is obtained.