On the coarse geometry of James spaces

In this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space $\mathcal{J}$ nor into its dual ${\mathcal{J}}^{\ast }$. It is a particular case of a more general result on the non equi-coarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic stucture. This allows us to exhibit a coarse invariant for Banach spaces, namely the non equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property $\mathcal{Q}$ of Kalton. We conclude with a remark on the coarse geometry of the James tree space $\mathcal{J}\mathcal{T}$ and of its predual.