Coarse embedding into uniformly convex Banach space

TL;DR

This paper investigates conditions under which coarse embeddings into Banach spaces are preserved when combining metric spaces, with a focus on groups hyperbolic relative to subgroups and embeddings into uniformly convex Banach spaces.

Contribution

It establishes that coarse embeddability into uniformly convex Banach spaces is preserved for certain group subsets when the group is hyperbolic relative to a subgroup that admits such an embedding.

Findings

Coarse embedding property can be preserved under unions of metric spaces.

If a subgroup admits a coarse embedding into a uniformly convex Banach space, so do certain associated group subsets.

The results apply to groups strongly relatively hyperbolic to such subgroups.

Abstract

In this paper, we study the coarse embedding into Banach space. We proved that under certain conditions, the property of embedding into Banach space can be preserved under taking the union the metric spaces. For a group $G$ strongly relative hyperbolic to a subgroup $H$, we proved that if $H$ admits a coarse embedding into a uniformly convex Banach space, so is $B(n)=\{g\in G|\abs{g}_{S\cup\mathscr{H}}\leq n\}$.