  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.

Metric Geometry