Classifying spaces for families of subgroups of 8-located groups

We investigate the structure of the minimal displacement set in $8$-located complexes with the SD’-property. We show that such set embeds isometrically into the complex. Since $8$-location and simple connectivity imply Gromov hyperbolicity, the minimal displacement set in such complex is systolic. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on an $8$-located complex with the SD’-property.