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

群是数学家用来抽象地描述对称性概念的工具。由于数学和基础科学中的许多结构都是非常对称的，因此群在这些领域的应用比比皆是。从粒子物理学的预测到纠错码，使光盘即使在脏或刮伤的情况下也能再现清晰的声音，许多科学领域都利用了群论。贯穿南安普顿进行的大部分研究的一个主题是对几何物体或空间的研究，其对称性体现了给定的群。例如，晶体的对称性可以用群来理解，即所谓的晶体群。晶体群是一个允许有限维模型的群的例子，该模型用于对真作用的分类空间进行分类，它通过与著名的鲍姆-康纳斯猜想的联系而引人注目。在这个项目中，我们将研究一些较弱的，代数上，或者更精确地说，同调，定义的不变量特征群承认一个有限维模型的分类空间适当的行动。这项工作将使我们能够在回答该领域一些长期存在的问题方面取得进展。特别地，我们将集中于具有无界挠率的群，这些问题的答案仍然是未知的。作为一个起点，我们将集中在分支集团，一个相对较新的和非常活跃的领域，几何群论。自Gromov于1991年提出拟等距不变量以来，它的研究已成为纯数学中一个非常重要而活跃的领域。目的是了解哪些同调性质是大尺度几何性质，即由拟等距保持。最近，Y. Shalom和R.绍尔介绍了方法从同调代数和代表性理论的领域证明准等距不变性的各种同调有限性条件。该项目的一个目的是，通过扩展他们的工作，了解上述同调不变量。

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.