## Infinite systolic groups are not torsion

### Volume 153 / 2018

Colloquium Mathematicum 153 (2018), 169-194
MSC: 20F65, 05E45, 05E18.
DOI: 10.4064/cm6982-6-2017
Published online: 30 April 2018

#### 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 $k \geq 7$ 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.