Proper actions on $\ell^p$ spaces for relatively hyperbolic groups

TL;DR

This paper proves that relatively hyperbolic groups act properly on uniformly convex Banach spaces if their peripheral subgroups do, extending the class of groups known to have such actions.

Contribution

It establishes that relative hyperbolicity combined with proper actions on uniformly convex spaces in subgroups implies the same for the entire group.

Findings

Relatively hyperbolic groups act properly on uniformly convex Banach spaces.

Peripheral subgroups' actions extend to the whole group.

Provides new examples of groups with proper affine isometric actions.

Abstract

We show that for any group $G$ that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then $G$ acts properly on a uniformly convex Banach space as well.