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Geometric methods in cohomology of soluble groups and their generalisations

可溶群上同调的几何方法及其推广

基本信息

批准号：
EP/F045395/1
负责人：
Brita Nucinkis
金额：
$ 2.07万
依托单位：
University of Southampton
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2008
资助国家：
英国
起止时间：
2008 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FF045395%2F1
关键词：
Geometric methods cohomology soluble groups

项目摘要

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.

群是数学家用来抽象地捕捉对称性概念的工具。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一些群论。贯穿在南安普顿进行的大部分研究的一个主题是几何对象或空间的研究，其对称性体现了给定的群体。例如，晶体的对称性已经用群，即所谓的晶体群得到了很好的理解。晶体学群是可溶群的例子，我们将使用几何方法研究一类群。由Bieri，Neumann和Strebel开发的Sigma-不变量是非常强大的几何工具，可以提供有关群的同调有限性条件的信息。确定的行为下传递到中心化的有限子群的这些西格玛-不变量将给出答案的一些重要问题，从代数拓扑。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。其目的是了解哪些代数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表示论的面积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，回答几个主要的问题，连接同源性和准等距。