Warped cones and spectral gaps

TL;DR

This paper demonstrates that warped cones over actions with spectral gaps cannot be coarsely embedded into many Banach spaces, including $L_p$-spaces and spaces with non-trivial type, revealing limitations in their geometric embeddings.

Contribution

It establishes new non-embeddability results for warped cones over actions with spectral gaps into various Banach spaces, extending understanding of their geometric properties.

Findings

Warped cones over actions with spectral gaps do not embed into $L_p$-spaces.

Warped cones over certain group actions cannot embed into any Banach space with non-trivial type.

Existence of warped cones with non-embeddability properties related to spectral gaps.

Abstract

We show that warped cones over actions with spectral gaps do not embed coarsely into large classes of Banach spaces. In particular, there exist warped cones over actions of the free group that do not embed coarsely into $L_p$-spaces and there are warped cones over discrete group actions that do not embed into any Banach space with non-trivial type.