This is an announcement for the paper “Coarse embeddings into superstable spaces” by Bruno de Mendonça Bragahttps://arxiv.org/find/math/1/au:+Braga_B/0/1/0/all/0/1, Andrew Swifthttps://arxiv.org/find/math/1/au:+Swift_A/0/1/0/all/0/1.

Abstract: Krivine and Maurey proved in 1981 that every stable Banach space contains almost isometric copies of $\ell_p$, for some $p\in[1, \infty)$. In 1983, Raynaud showed that if a Banach space uniformly embeds into a superstable Banach space, then $X$ must contain an isomorphic copy of $\ell_p$, for some $p\in[1, \infty)$. 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 $\ell_p$, for some $p\in[1, \infty)$. In particular, we obtain that there exist reflexive Banach spaces which do not coarsely embed into any superstable Banach space.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.04468