On Bredon (co-)homological dimensions of groups

Abstract

The objects of interest in this thesis are classifying spaces EFG for discrete groups G with stabilisers in a given family F of subgroups of G. The main focus of this thesis lies in the family Fvc(G) of virtually cyclic subgroups of G. A classifying space for this specific family is denoted by EG. It has a prominent appearance in the Farrell–Jones Conjecture. Understanding the finiteness properties of EG is important for solving the conjecture. This thesis aims to contribute to answering the following question for a group G: what is the minimal dimension a model for EG can have? One way to attack this question is using methods in homological algebra. The natural choice for a cohomology theory to study G-CW-complexes with stabilisers in a given family F is known as Bredon cohomology. It is the study of cohomology in the category of OFG-modules. This category relates to models for EFG in the same way as the category of G-modules relates to the study of universal covers of Eilenberg–Mac Lane spaces K(G; 1). In this thesis we study Bredon (co-)homological dimensions of groups. A major part of this thesis is devoted to collect existing homological machinery needed to study these dimensions for arbitrary families F. We contribute to this collection. After this we turn our attention to the specific case of F = Fvc(G). We derive a geometric method for obtaining a lower bound for the Bredon (co-)homological dimension of a group G for a general family F, and subsequently show how to exploit this method in various cases for F = Fvc(G). Furthermore we construct model for EG in the case that G belongs to a certain class of infinite cyclic extensions of a group B and that a model for EB is known. We give bounds on the dimensions of these models. Moreover, we use this construction to give a concrete model for EG, where G is a soluble Baumslag–Solitar group. Using this model we are able to determine the exact Bredon (co-)homological dimensions of these groups. The thesis concludes with the study of groups G of low Bredon dimension for the family Fvc(G) and we give a classification of countable, torsion-free, soluble groups which admit a tree as a model for EG

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