Geometric study of infinite discrete groups

本课题的目的是从几何的角度研究无限离散群的若干问题。“几何群论”起源于格罗莫夫的早期工作，近15年来主要在美国和欧洲得到发展。它是一个动态的领域，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，日本在这方面的研究还不多，在过去的三年里，我们不仅进行了研究，而且试图为日本在这一领域的研究活动奠定基础，我们的研究主题之一就是在CAT(0)空间上的群的等距行为。“CAT(0)空间”的概念是由Gromov引入测地度量空间的，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，适当地找到G按等距作用的度量空间X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。找到这样的极小维度的X将是有趣的。在这个方向上，布雷迪-克里斯普已经做了一项工作。我们发展了他们的工作，找到了他们问题的答案，找到了新的例子，并在Fujiwara，Koji，Shioya，Takashi，Yamagata，Saeko的论文《CAT(0)空间和CAT(0)维的抛物线等距》中提出了进一步的问题。《代数地理》4(2004)，861-892

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