Infinite systolic groups are not torsion

We study k-systolic complexes introduced by T. Januszkiewicz and J. \'Swi\k{a}tkowski, which are simply connected simplicial complexes of simplicial nonpositive curvature. Using techniques of filling diagrams we prove that for k > 6 the 1-skeleton of a k-systolic complex is Gromov hyperbolic. We give an elementary proof of so-called Projection Lemma, which implies contractibility of 6-systolic complexes. We also prove that an infinite group acting geometrically on a 6-systolic complex is not torsion.