Fibred cofinitely-coarse embeddability of box families and proper isometric affine actions on uniformly convex Banach spaces

Abstract

In this paper we show that a countable, residually amenable group admits a proper isometric affine action on some uniformly convex Banach space if and only if one (or equivalently, all) of its box families admits a fibred cofinitely-coarse embedding into some uniformly convex Banach space.

Uniformly bounded fibred coarse embeddability and uniformly bounded a-T-menability

“…This result was also proved by M. Finn-Sell in [8] using groupoid method. The above statements had been generalized to the case of L p -spaces by S. Arnt [1], to the case of uniformly convex Banach spaces by G. Li and X. Wang [14]. If we consider warped cones (see Definition 4.12) rather than box spaces, there are some similar results obtained by J. Roe [21], D. Sawicki and J. Wu [22], Q. Wang and Z. Wang [25].…”