Infinite systolic groups are not torsion

Abstract

We study $k$-systolic complexes introduced by T. Januszkiewicz and J. Świątkowski, 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.

arXiv commentVersion 3, 27 pages, 10 figures. Major revision. Proof of Theorem 1.2 corrected, proof of Theorem 4.3 simplified, a reference to an alternative proof of Theorem 7.4 added. Several definitions and lemmas adjusted and few typos removed. Language and exposition improved. Version very similar to the published version