Fibred coarse embeddability of box spaces and proper isometric affine actions on $L^p$ spaces

TL;DR

This paper establishes a deep connection between the geometric property of fibred coarse embeddability of box spaces and the algebraic property of a group having a proper isometric affine action on an $L^p$ space, providing new insights into geometric group theory.

Contribution

It proves that a finitely generated, residually finite group has property $PL^p$ if and only if its box space admits a fibred coarse embedding into an $L^p$ space, linking geometric and algebraic properties.

Findings

Fibred coarse embeddability of box spaces characterizes property $PL^p$ for groups.

Coarse embeddability of a box space into $L^p$ implies the group has property $PL^p$.

The necessary condition for a group to have property $PL^p$ is established.

Abstract

We show the necessary part of the following theorem : a finitely generated, residually finite group has property $PL^p$ (i.e. it admits a proper isometric affine action on some $L^p$ space) if, and only if, one (or equivalently, all) of its box spaces admits a fibred coarse embedding into some $L^p$ space. We also prove that coarse embeddability of a box space of a group into a $L^p$ space implies property $PL^p$ for this group.