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.