Coarse embedding into uniformly convex Banach space

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 $B(n)=\{g\in G|\abs{g}_{S\cup\mathscr{H}}\leq n\}$.