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

TL;DR

This paper establishes a deep connection between the geometric property of fibred cofinitely-coarse embeddability of box families of groups and the existence of proper isometric affine actions on uniformly convex Banach spaces, providing a characterization for residually amenable groups.

Contribution

It proves an equivalence between fibred cofinitely-coarse embeddability of box families and proper affine actions on uniformly convex Banach spaces for residually amenable groups.

Findings

Residually amenable groups admit proper affine actions iff their box families embed fibred cofinitely-coarsely.

Characterizes geometric group properties via Banach space actions.

Links coarse geometric properties with functional analysis in group theory.

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.