Filling invariants of systolic complexes and groups

Abstract

Systolic complexes are simplicial analogues of nonpositively curved spaces. Their theory seems to be largely parallel to that of CAT(0) cubical complexes.

We study the filling radius of spherical cycles in systolic complexes, and obtain several corollaries. We show that a systolic group can not contain the fundamental group of a nonpositively curved Riemannian manifold of dimension strictly greater than 2, although there exist word hyperbolic systolic groups of arbitrary cohomological dimension.

We show that if a systolic group splits as a direct product, then both factors are virtually free. We also show that systolic groups satisfy linear isoperimetric inequality in dimension 2.