Coarse embeddings into superstable spaces

Krivine and Maurey proved in 1981 that every stable Banach space contains almost isometric copies of ${\mathrm{\ell }}_p$, for some $p\in [1,\mathrm{\infty })$. In 1983, Raynaud showed that if a Banach space uniformly embeds into a superstable Banach space, then $X$ must contain an isomorphic copy of ${\mathrm{\ell }}_p$, for some $p\in [1,\mathrm{\infty })$. In these notes, we show that if a Banach space coarsely embeds into a superstable Banach space, then $X$ has a spreading model isomorphic to ${\mathrm{\ell }}_p$, for some $p\in [1,\mathrm{\infty })$. In particular, we obtain that there exist reflexive Banach spaces which do not coarsely embed into any superstable Banach space.