Geometric methods in cohomology of soluble groups and their generalisations

群是数学家用来抽象地捕捉对称概念的工具。由于数学和基础科学中的许多结构都是非常对称的，基团在这些领域的应用很多。从粒子物理学的预测，到使光盘即使在脏的或有划痕的情况下也能重现清晰声音的纠错码，许多科学领域都利用了一些群论。在南安普敦进行的大部分研究中，贯穿其中的一个主题是对几何对象或空间的研究，几何对象或空间的对称性体现了给定的群体。例如，使用基团，即所谓的结晶学基团，就可以很好地理解晶体的对称性。晶体基团是可溶基团的例子，这是我们将用几何方法研究的一类基团。由Bieri，Neumann和Strebel发展的Sigma不变量是非常强大的几何工具，它给出了关于群的同调有限条件的信息。确定这些Sigma不变量在传递到有限子群的中心子下的行为将给出代数拓扑学中的一些重要问题的答案。拟等距不变量的研究由格罗莫夫于1991年首创，现已成为纯数学中一个非常重要和活跃的领域。其目的是了解有限生成群的哪些代数性质是大规模几何性质，即由拟等距保持的。最近，Y.Shalom和R.Sauer的开创性工作将同调代数和表示论的方法引入到证明各种同调有限条件的拟等距不变性的领域。该项目的一个目标是，通过扩展他们的工作，回答几个将同调和准等距联系起来的主要问题。

A group is the mathematician's tool to capture the notion of symmetry in 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 soluble groups, a class of groups we will be in- vestigating using geometric methods. The Sigma-invariants developed by Bieri, Neumann and Strebel are very powerful geometric tools giving information about homological finiteness conditions of a group. Determining the behaviour of these Sigma-invariants under passing to centralisers of finite subgroups will give answers to some important questions from algebraic topology. 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 algebraic properties of finitely generated groups are large scale geometric properties, i.e. are pre- served 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 answer several of the main questions linking homology and quasi-isometry.