Geometric Group Theory: Group Boundaries and the Torus Theorem

几何群论：群边界和环面定理

• 批准号: 9803461
• 负责人: Eric Swenson
• 金额: $ 5.8万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803461&HistoricalAwards=false
• 关键词: Geometric; Group; Theory; Boundaries; Torus

9803461 Swenson First, Swenson will continue his collaboration with Martin Dunwoody on the algebraic torus theorem. Much of the work done in the field of geometric group theory has been generalizing results from 3-manifold theory to the setting of finitely generated groups. The first result to generalize was the sphere theorem, which was generalized by Stallings to the Ends Theorem. When the most general versions of the annulus and torus theorem for three-manifolds were proven by Scott in 1980, he conjectured that there should be a purely algebraic analog to these theorems. The first step in that direction was a proof of the algebraic annulus theorem for word hyperbolic groups by Scott and Swarup in 1995. The algebraic annulus theorem for finitely generated groups was proven by Dunwoody and Swenson in 1996. These two will extend this result to an n-dimensional algebraic torus theorem for finitely generated groups. In addition, Swenson will continue his investigations into the boundaries of groups. The study of group boundaries was begun in the theory of Kleinian and Fuchsian groups. The first person to apply these ideas to abstract groups was Gromov. He defined the visual boundary both of a word-hyperbolic group, and of a CAT(0) group. It was shown by Bestvina and Mess that these boundaries carry considerable cohomological information about the group. Swenson has two different projects in this area. First, he wishes to determine the relationship between the local and global homology in the visual boundary of a word-hyperbolic group. Bestvina began the work in this area, after which Swenson made a contribution. There are, however, certain important questions that remain open. The second project concerns the existence (or non-existence) of cut points in these boundaries. It has been shown by Bowditch and Swarup that the visual boundary of a word-hyperbolic group cannot contain a cut-point. Swenson will try to "extend" this result to the nonpositive ly curved or CAT(0) setting. He will also look for consequences of boundaries' not having cut-points (semistability at infinity, for example). Years ago it was said that infinite group theory consisted entirely of counter-examples. Geometric group theory has done much to remedy this situation. Thurston's work towards the classification of all 3-manifolds, spaces which look locally like 3-space, was the origin of geometric group theory. Z. He, Cannon, and others noticed the amazing correlation between the group theoretic properties of the fundamental group of a geometric 3-manifold and the geometric properties of its universal cover. Much of the work done in geometric group theory has been to translate techniques and sometimes even theorems from the setting of 3-manifolds to the seemingly much more general setting of simply connected 2-complexes, spaces formed by gluing triangles along their edges (often with more than two triangles sharing a common edge). The torus theorem for 3-manifolds says that any time there is an essential map of a 2-torus (an inner tube) into a 3-manifold M, either M is very simple and well understood, or there is an embedded 2-torus along which M can be cut to yield simpler pieces. Swenson is working with Martin Dunwoody to extend this result to apply to any essential map of a torus of any dimension into a simply connected 2-complex. In the second part of the project, Swenson will investigate non-positively curved 2-complexes, using techniques transferred over from differential geometry by Gromov and others. ***

小行星9803461 首先，斯文森将继续他的合作与马丁邓伍迪的代数环面定理。 在几何群论领域中所做的大量工作是将3-流形理论的结果推广到双生成群的设置。 推广的第一个结果是球面定理，Stallings将其推广为端点定理。 1980年，当斯科特证明了三流形的环和环面定理的最一般版本时，他推测这些定理应该有一个纯粹的代数类似物。 第一步，在这个方向上是一个证明的代数环定理字双曲群斯科特和Swarup在1995年。 代数环定理是由Dunwoody和Swenson在1996年证明的。 这两个将这个结果推广到n维代数环面定理的n-生成群。 此外，斯文森将继续调查群体的边界。 群边界的研究始于克莱因群和富克斯群的理论。 第一个将这些思想应用于抽象群体的人是格罗莫夫。 他定义了单词双曲群和CAT（0）群的视觉边界。 Besteland和Mess指出，这些边界携带着关于群的大量上同调信息。 斯文森在这个领域有两个不同的项目。 首先，他希望确定的局部和整体之间的关系，在视觉边界的词双曲组的同调。 Besteland开始了这方面的工作，之后Swenson做出了贡献。 然而，仍有一些重要问题有待解决。 第二个项目涉及这些边界中是否存在（或不存在）分界点。 Bowditch和Swarup已经证明，词双曲群的视觉边界不能包含一个截点。 Swenson将尝试将此结果“扩展”到非正曲线或CAT（0）设置。 他还将寻找边界没有截点的后果（例如，无穷远处的半稳定性）。 几年前，有人说无限群理论完全由反例组成。 几何群论做了很多工作来纠正这种情况。 瑟斯顿的工作对分类的所有3-流形，空间看起来当地像3-空间，是起源的几何群论。 Z.他，坎农，和其他人注意到惊人的相关性群论性质的基本组的几何3流形和几何性质的普遍覆盖。 在几何群论中所做的许多工作都是将技巧，有时甚至是定理从3-流形的设置转换到似乎更一般的单连通2-复形的设置，即由沿着它们的边粘合三角形形成的空间（通常有两个以上的三角形共享一条公共边）。 三维流形的环面定理说，任何时候都有一个2-环面（一个内管）到三维流形M的本质映射，要么M非常简单并且很容易理解，要么有一个嵌入的2-环面沿着，M可以被切割成更简单的部分。 斯文森正在与马丁邓伍迪扩展这一结果适用于任何基本地图的任何层面的环面到一个简单的连接2复杂。 在该项目的第二部分，斯文森将调查非积极弯曲的2-复合物，使用的技术转移到微分几何的格罗莫夫和其他人。 ***