Thin actions on CAT(0) spaces

We study groups of isometries of packed, geodesically complete, CAT$(0)$-spaces for which the systole at every point is smaller than a universal constant depending only on the packing, deducing strong rigidity results. We show that if a space as above has some negative curvature behaviour then it cannot support a thin action: this generalizes the classical Margulis Lemma to a broader class of spaces.