Warped cones over profinite completions

TL;DR

This paper constructs warped cones that are coarsely embeddable into Hilbert space but lack property A, providing new examples that challenge previous assumptions about these properties.

Contribution

It introduces the first warped cones combining coarse embeddability with the absence of property A and generalizes criteria for property A and embeddability.

Findings

Warped cones over profinite completions are coarsely embeddable but lack property A.

Property A of warped cones over profinite groups is equivalent to group amenability.

Constructed warped cones over manifolds with embedded expanders.

Abstract

We construct metric spaces that do not have property A yet are coarsely embeddable into the Hilbert space. Our examples are so called warped cones, which were introduced by J. Roe to serve as examples of spaces non-embeddable into a Hilbert space and with or without property A. The construction provides the first examples of warped cones combining coarse embeddability and lack of property A. We also construct warped cones over manifolds with isometrically embedded expanders and generalise Roe's criteria for the lack of property A or coarse embeddability of a warped cone. Along the way, it is proven that property A of the warped cone over a profinite completion is equivalent to amenability of the group. In the appendix we solve a problem of Nowak regarding his examples of spaces with similar properties.