Note on distortion and Bourgain ${\ell }_1$ index

Abstract

The relation between different notions measuring proximity to ${\ell }_1$ and distortability of a Banach space is studied. The main result states that a Banach space, whose all subspaces have Bourgain ${\ell }_1$ index greater than ${\omega }^{\alpha }$, $\alpha <{\omega }_1$, contains either an arbitrary distortable subspace or an ${\ell }_1^{\alpha }$-asymptotic subspace.

14 pages, remarks and references added