Infinite systolic groups are not torsion

We study $k$-systolic complexes introduced by T. Januszkiewicz and J. '{S}wi\k{a}tkowski, which are simply connected simplicial complexes of simplicial nonpositive curvature. Using techniques of filling diagrams we prove that for $k\ge 7$ the $1$-skeleton of a $k$-systolic complex is Gromov hyperbolic. We give an elementary proof of the so-called Projection Lemma, which implies contractibility of $6$-systolic complexes. We also present a new proof of the fact that an infinite group acting geometrically on a $6$-systolic complex is not torsion.