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

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