# Straightening warped cones

## TL;DR

This paper establishes converses to known results about warped cones, showing that certain properties imply group amenability and fibred coarse embeddings, with examples demonstrating the necessity of freeness in these conditions.

## Contribution

It proves the converses to Roe's results on warped cones, extending the equivalence to other Banach spaces and providing examples that highlight the importance of the freeness assumption.

## Key findings

- Warped cones from free actions of a-T-menable groups admit fibred coarse embeddings into Hilbert spaces.
- Warped cones with property A from free actions imply the group is amenable.
- Examples show freeness is necessary for these properties to hold.

## Abstract

We provide the converses to two results of J. Roe (Geom. Topol. 2005): first, the warped cone associated to a free action of an a-T-menable group admits a fibred coarse embedding into a Hilbert space, and second, a free action yielding a warped cone with property A must be amenable. We construct examples showing that in both cases the freeness assumption is necessary. The first equivalence is obtained also for other classes of Banach spaces, in particular for $L^p$-spaces.