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We study the theory of convergence for CAT (0) -lattices (that is groups acting geometrically on proper, geodesically complete CAT (0) -spaces) and their quotients (CAT (0) -orbispaces). We describe some splitting and collapsing phenomena, explaining precisely how these action can degenerate to a possibly non-discrete limit action. Finally, we prove a compactness theorem for the class of compact CAT (0) -homology orbifolds, and some applications: an isolation result for flat orbispaces and an entropy-pinching theorem.
Cavallucci et al. (Tue,) studied this question.