Let $h$ be a hyperbolic isometry of a systolic complex (a simply connected complex of simplicial non-positive curvature). In joint work with D. Osajda we show that the minimal displacement set of $h$ decomposes up to quasi-isometry as the product of a tree and the real line. From this we deduce the following two corollaries. For a group $G$ acting properly on a systolic complex we construct a finite dimensional model for the classifying space EG. If the action is additionally cocompact, we prove that the centralizer of any hyperbolic element is virtually free-by-cyclic. Before outlining these results I shall give some background on classifying spaces for families and systolic complexes.