权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

The complex hyperbolic structures on the configuration spaces of points on the sphere and surface subgroup of mapping class groups

映射类群球面子群上点位形空间上的复双曲结构

基本信息

批准号：
18540085
负责人：
YAMASHITA Yasushi
金额：
$ 0.74万
依托单位：
Nara Women's University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540085/
关键词：
Configuration space Hyperbolic geometry Kleinian groups オートマティック群

项目摘要

(1) Structures of non-hyperbolic automatic groups(Joint work with Y. Nakagawa, M. Tamura)Let G be a finitely presented group. If G contains a Z + Z subgroup, then it is well known that G cannot be word hyperbolic. A natural question is that "is Z + Z the only obstruction for a finitely presented group to be word hyperbolic?" In other words, "if G does not contain any Z + Z subgroups, is it word hyperbolic?" Baumslag-Solitar groups are counter examples to this question. Thus it would be better to restrict our attention to some good class of groups. Here we focus on automatic groups. Note that Baumslag-Solitar groups are not automatic. Our problem is indicated in the list of open problems and attributed to Gersten. We call this problem "Gersten's problem".Recall that the class of all automatic groups contains the class of all hyperbolic groups, all virtually abelian groups and all geometrically finite hyperbolic groups. A geometrically finite hyperbolic group is, in some sense, similar to hyperbolic groups, but it might contain a Z + Z subgroup. Thus the class of automatic groups is a nice target to consider the question mentioned before.We define the notion of "n-tracks of length n", which suggests a clue of the existence of Z + Z subgroup and shows its existence in every non-hyperbolic automatic groups with mild conditions.(2) The character variety of one-holed torus(Joint work with S.P. Tan)The quasifuchsian space of punctured torus groups is deeply studied by many people and some of the major conjectures on them are solved in the last decade. But, for general "one-holed" cases, not much is known. In this study, we produced computer software to investigate the character variety of one holed torus and were able to find many interasting phenomena

(1) 非双曲自动群的结构(与Y. Nakakawa, M. Tamura 合作) 设G 为有限呈现群。如果 G 包含 Z + Z 子群，则众所周知 G 不能是字双曲线。一个自然的问题是“Z + Z 是有限呈现群成为词双曲线的唯一障碍吗？”换句话说，“如果 G 不包含任何 Z + Z 子群，那么它是双曲词吗？” Baumslag-Solitar 群是这个问题的反例。因此，最好将我们的注意力限制在一些好的群体上。这里我们重点关注自动组。请注意，Baumslag-Solitar 群不是自动的。我们的问题已在未决问题列表中指出并归因于 Gersten。我们称这个问题为“格斯滕问题”。回想一下，所有自动群的类包含所有双曲群、所有虚拟交换群和所有几何有限双曲群的类。在某种意义上，几何有限双曲群与双曲群类似，但它可能包含 Z + Z 子群。因此，自动群的类是考虑前面提到的问题的一个很好的目标。我们定义了“长度为n的n轨道”的概念，它提出了Z + Z子群存在的线索，并表明它存在于每个具有温和条件的非双曲自动群中。（2）单孔环面的特征变化（与S.P. Tan联合工作）穿孔环面群的拟福克空间被许多人深入研究。人们以及关于他们的一些主要猜想在过去十年中得到了解决。但是，对于一般的“单孔”情况，人们知之甚少。在这项研究中，我们制作了计算机软件来研究单孔环面的特征变化，并发现了许多有趣的现象