Superexpanders from group actions on compact manifolds

TL;DR

This paper constructs new examples of superexpanders from group actions on compact manifolds that do not embed into Banach spaces with nontrivial type, and establishes their distinctness from Lafforgue expanders.

Contribution

It introduces infinitely many superexpanders from warped cones that are not Lafforgue expanders and proves their non-embeddability into Banach spaces with nontrivial type.

Findings

New superexpanders do not coarsely embed into Banach spaces with nontrivial type.

These superexpanders are not coarsely equivalent to Lafforgue expanders.

A quasi-isometric rigidity result for warped cones is established.

Abstract

It is known that the expanders arising as increasing sequences of level sets of warped cones, as introduced by the second-named author, do not coarsely embed into a Banach space as soon as the corresponding warped cone does not coarsely embed into this Banach space. Combining this with non-embeddability results for warped cones by Nowak and Sawicki, which relate the non-embeddability of a warped cone to a spectral gap property of the underlying action, we provide new examples of expanders that do not coarsely embed into any Banach space with nontrivial type. Moreover, we prove that these expanders are not coarsely equivalent to a Lafforgue expander. In particular, we provide infinitely many coarsely distinct superexpanders that are not Lafforgue expanders. In addition, we prove a quasi-isometric rigidity result for warped cones.