  We determine the large scale geometry of the minimal displacement set of a
hyperbolic isometry of a systolic complex. As a consequence, we describe the
centraliser of such an isometry in a systolic group. Using these results, we
construct a low-dimensional classifying space for the family of virtually
cyclic subgroups of a group acting properly on a systolic complex. Its
dimension coincides with the topological dimension of the complex if the latter
is at least four. We show that graphical small cancellation complexes are
classifying spaces for proper actions and that the groups acting on them
properly admit three-dimensional classifying spaces with virtually cyclic
stabilisers. This is achieved by constructing a systolic complex equivariantly
homotopy equivalent to a graphical small cancellation complex. On the way we
develop a systematic approach to graphical small cancellation complexes.
Finally, we construct low-dimensional models for the family of virtually
abelian subgroups for systolic, graphical small cancellation, and some CAT(0)
groups.
