Convergence and collapsing of CAT(0)-lattices

We study the theory of convergence for CAT(0)-lattices (that is groups Γ 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 the actions can degenerate to a possibly non-discrete limit action, and prove a compactness theorem for the class of compact CAT(0)-homology orbifolds. Finally, as an application of this theory, we prove an isolation result for flat orbispaces and an entropy-pinching theorem.