Geometric study of infinite discrete groups

本课题的目的是从几何的角度研究无穷离散群的几个问题。“几何群论”起源于格罗莫夫的开创性工作，近15年来主要在美国和欧洲得到发展。这是一个动态的领域，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的理论。遗憾的是，日本在这方面的活动还不多。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群在CAT(0)空间上的等距作用。“CAT(0)空间”的概念由Gromov引入到测地线度量空间中，作为完全单连通黎曼流形（称为“Hadamard流形”）的推广。给定一个离散群G，找到G作用于其上的度量空间X是非常重要和有用的。一个经典的例子是晶格子群在李群的对称空间上的作用。找到这样的最小维的X是很有趣的。在这个方向上，布雷迪-克里斯普（Brady-Crisp）做过一项研究。我们发展了他们的工作，找到了他们问题的答案，发现了新的例子，并在论文“CAT(0)空间和CAT(0)维度的抛物线等距”中提出了进一步的问题，Fujiwara, Koji；盐谷，高桥；山形,Saeko。Algebr.Geom.Topol。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