Actions on Products of CAT(-1) Spaces

We show that for X a proper CAT(-1) space there is a maximal open subset of the horofunction compactification of X× X , with respect to the maximum metric, that compactifies the diagonal action of an infinite quasi-convex group of the isometries of X. We also consider the product action of two quasi-convex representations of an infinite hyperbolic group on the product of two different proper CAT(-1) spaces.