Horoboundaries of coarsely convex spaces
Ikkei Sato
TL;DR
This work studies horoboundaries of coarsely convex spaces and their relationship with ideal boundaries. It introduces a cone metric $d_c$ on a coarsely convex space $(X,d)$ and defines the horoboundary $\partial^c_h X$ and the reduced horoboundary $\partial^c_h X/\sim$. It then proves that the horoboundary of the open cone $\mathcal{O}\partial_o X$ is homeomorphic to the ideal boundary $\partial_o X$, and, for geodesic coarsely convex spaces, the reduced horoboundary $\partial^c_h X/\sim$ is homeomorphic to $\partial_o X$. These results generalize Andreev’s cone-metric correspondence from Busemann spaces to the broader class of coarsely convex spaces, bridging horoboundary and ideal boundary theories in coarse geometry.
Abstract
A horoboundary is one of the attempts to compactify metric spaces, and is constructed using continuous functions on metric spaces. It is a concept that includes global information of metric spaces, and its correspondence with an ideal boundary constructed using geodesics has been studied in nonpositive curvature spaces such as CAT(0) spaces and geodesic Gromov hyperbolic spaces. We will introduce a certain correspondence between the horoboundary and the ideal boundary of coarsely convex spaces, which can be regarded as a generalization of spaces of nonpositive curvature.