Classifying spaces for chains of families of subgroups

Abstract

This thesis concerns the study of the Bredon cohomological and geometric dimensions of a discrete group $G$ with respect to a family $\mathfrak{F}$ of subgroups of $G$. With that purpose, we focus on building finite-dimensional models for $\operatorname{E}_{\mathfrak{F}} \left( G \right)$. The cases of the family $\mathfrak{Fin}$ of finite subgroups of a group and the family $\mathfrak{VC}$ of virtually cyclic subgroups of a group have been widely studied and many tools have been developed to relate the classifying spaces for $\mathfrak{VC}$ with those for $\mathfrak{Fin}$. Given a discrete group $G$ and an ascending chain $\mathfrak{F}_0 \subseteq \mathfrak{F}_1 \subseteq \ldots \subseteq \mathfrak{F}_n \subseteq \ldots$ of families of subgroups of $G$, we provide a recursive methodology to build models for $\operatorname{E}_{\mathfrak{F}_r} \left( G \right)$ and give certain conditions under which the models obtained are finite-dimensional. We provide upper bounds for both the Bredon cohomological and geometric dimensions of $G$ with respect to the families $\left(\mathfrak{F}_r\right)_{r\in\mathbb{N}}$ utilising the classifying spaces obtained. We consider then the families $\mathfrak{H}_r$ of virtually polycyclic subgroups of Hirsch length less than or equal to $r$, for $r\in\mathbb{N}$. We apply the results obtained for chains of families of subgroups to the chain $\mathfrak{H}_0 \subseteq \mathfrak{H}_1 \subseteq \ldots$ for an arbitrary virtually polycyclic group $G$, proving that the corresponding Bredon dimensions are both bounded above by $h(G) + r$, where $h(G)$ is the Hirsch length of $G$. Finally, we give similar results for the same chain of families of subgroups and an arbitrary locally virtually polycyclic group as the ambient group, obtaining in this case the upper bound $h(G) + r + 1$.