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

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