Classifying spaces for chains of families of subgroups

TL;DR

This thesis develops recursive methods to construct finite-dimensional models for classifying spaces of groups with respect to chains of subgroup families, providing bounds on their Bredon cohomological and geometric dimensions.

Contribution

It introduces a recursive approach to build models for classifying spaces for chains of subgroup families and establishes bounds on their dimensions, especially for virtually polycyclic groups.

Findings

Provided recursive methodology for models of classifying spaces.

Established upper bounds for Bredon dimensions based on subgroup chains.

Applied results to virtually polycyclic groups, bounding dimensions by Hirsch length.

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 $h(G) + r + 1$.