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Geometric study of infinite discrete groups

无限离散群的几何研究

基本信息

批准号：
14540055
负责人：
FUJIWARA Koji
金额：
$ 2.62万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540055/
关键词：
Discrete groups CAT(0) spaces 有界コホモロジー写像類群双曲群

项目摘要

The goal of this project is to study several problems on infinite discrete groups from the view point of geometry."Geometric group theory" has its root in the pionear work by Gromov, and has been developed mainly in USA and Europe in the last 15 years or so.It is a dynamic field where one can apply classical combinatorial group theory, hyperbolic geometry, low dimensional topology, and the theory of mapping class groups. Unfortunately, there has not been much activities of this field in Japan yet.During the three years, we not only conducted our research, but also tried to put a foundation of the research activities of this field in Japan.One of our research themes has been on isometric actions of group on CAT(0) spaces. The notion of "CAT(0) spaces" was introduced by Gromov to geodesic metric spaces as a generalization of complete, simply-connected Riemannian manifolds, called "Hadamard manifolds".Given a discrete group G, it is important and useful to find a metric space X on which G acts on by isometries, properly. One classical example is the action of a lattice subgroup in a Lie group on its symmetric space.It would be interesting to find such X of minimal dimensions. In this direction, there has been a work by Brady-Crisp. We developed their work and found an answer to their question, found new examples, and formulated further questions in the paper "Parabolic isometries of CAT(0) spaces and CAT(0) dimensions", Fujiwara, Koji ; Shioya, Takashi ; Yamagata, Saeko. Algebr.Geom.Topol.4(2004), 861-892

本项目的目标是从几何的角度研究无限离散群的几个问题。“几何群论”起源于Gromov的开创性工作，近15年来主要在美国和欧洲发展起来，是一个动态的领域，人们可以应用经典的组合群论、双曲几何、低维拓扑和映射类群理论。遗憾的是，这一领域在日本还没有太多的活动。在这三年中，我们不仅进行了我们的研究，而且试图为日本这一领域的研究活动奠定基础。我们的研究主题之一是CAT（0）空间上的群体等距行动。“CAT（0）空间”的概念是由Gromov引入到测地度量空间中的，它是完备的单连通黎曼流形的推广，称为“Hadamard流形”。给定一个离散群G，找到一个度量空间X，使G在X上等距地作用是非常重要和有用的。一个经典的例子是李群中的格子群在其对称空间上的作用，找到这样的最小维数的X是很有趣的。在这个方向上，有一个工作的Brady-Crisp。我们发展了他们的工作，找到了他们的问题的答案，发现了新的例子，并制定了进一步的问题，在文件“抛物线等距的CAT（0）空间和CAT（0）维”，藤原，浩二;盐屋，隆;山形，佐惠子。Algebr.Geom.Topol.4（2004），861-892