Compactification of a diagonal action on the product of CAT(-1) spaces

TL;DR

This paper studies the compactification of the diagonal action of a cocompact group on the product of CAT(-1) spaces using horofunction boundaries, providing a new geometric framework.

Contribution

It introduces a novel compactification method for diagonal group actions on product spaces of CAT(-1) spaces via horofunction boundaries.

Findings

Established a compactification framework for diagonal actions.

Characterized the boundary structure using horofunctions.

Enhanced understanding of group actions on non-compact spaces.

Abstract

Let $X$ be a proper, non-compact CAT(-1) space, and $\Gamma$ a discrete cocompact subgroup of the isometries of $X$. We compactify the diagonal action of $\Gamma$ on $X \times X$ considering a domain of the horofunction boundary with respect to the maximum metric.