# The coarse geometry of Tsirelson's space and applications

## TL;DR

This paper establishes a rigidity result for Banach spaces coarsely embeddable into Tsirelson's space, revealing structural constraints and disproving a conjecture about Hilbert space embeddings.

## Contribution

It introduces a new rigidity theorem for spaces coarsely embeddable into Tsirelson's space, with significant implications for coarse geometry and Banach space theory.

## Key findings

- Banach spaces coarsely embeddable into T* are reflexive and have spreading models isomorphic to c0
- T* does not coarsely contain c0 or ℓp for p in [1,∞)
- The conjecture that Hilbert space coarsely embeds into every Banach space is false

## Abstract

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 $T^*$. Every Banach space that is coarsely embeddable into $T^*$ must be reflexive and all its spreading models must be isomorphic to $c_0$. Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: $T^*$ does not coarsely contain $c_0$ nor $\ell_p$ for $p\in[1,\infty)$. 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 and taking values in $T^*$, and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to $c_0$. Also, a purely metric characterization of finite dimensionality is obtained.