Classifying spaces for chains of families of subgroups

Let F₀ ⊆ F₁ ⊆ ... ⊆ Fn ⊆ ...  be an ascending chain of families ofsubgroups of a discrete group G. Under certain conditions, it is possible tobuild classifying spaces for all the families in the chain recursively, utilisingthose for F₀ and other families that will be introduced throughout theprocess. The aim of this thesis is to provide such methodology and use itto give upper bounds for the respective Bredon dimensions. We then applythis construction process to families of virtually polycyclic subgroups.