Classifying spaces for proper actions and cohomological finiteness conditions of discrete groups.

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A group is the mathematician's tool to capture the notion of symmetry in the abstract. Since many structures in mathematics and the basic sciences are very symmetrical, applications of groups abound in these areas. From the predictions of particle physics to error correcting codes that enable compact discs to reproduce clear sound even when dirty or scratched, many areas of science utilise some group theory.One theme that runs throughout much of the research carried out in Southampton is the study of geometric objects, or spaces, whose symmetries embody the given group. The symmetry of crystals, for example, has been well understood using groups, the so called crystallographic groups.Crystallographic groups are examples of groups admitting a finite dimensional model for the classifying space for proper actions, which has come to prominence through its connection with the celebrated Baum-Connes conjecture. In this project we shall investigate some weaker, algebraically or, to be more precise, homologically, defined invariants characterising groups admitting a finite dimensional model for the classifying space for proper actions. This work will enable us to make progress in answering some long-standing conjectures in the field. In particular, we shall concentrate on groups having unbounded torsion, for which the answers to these questions are still unknown. As a starting point we will concentrate on Branch groups, a relatively new and extremely vibrant area in geometric group theory. Originated by Gromov in 1991, the study of quasi-isometry invariants has become a very important and active area in pure mathematics. The aim is to understand which ho- mological properties of finitely generated groups are large scale geometric properties, i.e. are preserved by quasi-isometry. Recently, the seminal work of Y. Shalom and R. Sauer introduced methods from homological algebra and representation theory to the area proving quasi-isometry invariance of various homological finiteness conditions. One aim of the project is, by extending their work, to understand the homological invariants mentioned above.