EAPSI: An Investigation of Closed Surfaces in 3-manifolds via Character Varieties

为了理解三维形状或三维流形的结构，研究三维流形中编码重要信息的某些表面通常会有所帮助。Culler-Shalen理论提供了最通用的工具，用于构建这样的表面，并已在许多著名的问题的解决办法。这些特殊的曲面自然分为两类：有边界的曲面和无边界的曲面。Culler-Shalen理论的大部分力量来自于对可以用这些工具构建的边界表面的深刻理解。另一方面，很少有人知道有关的表面没有边界。在本计画中，我们将从一个崭新的代数观点来研究Culler-Shalen理论中的无边界曲面。我们的工作将解决许多问题，关于Culler-Shalen理论本身，也将打开大门，将理论应用到更广泛的一类问题，3-流形。这项研究将与澳大利亚悉尼大学的Culler-Shalen理论和三维流形的主要专家Stephan Tillman博士合作进行。更具体地说，Culler-Shalen理论使用三维流形的特征簇的代数几何和Stallings的结构来构建流形中的基本曲面。A-多项式和Culler-Shalen范数都精确地确定了当流形的边界由单个环面组成时，哪些边界斜率以这种方式出现。然而，并不是每一个边界斜率检测Culler-Shalen理论。我们将解决一个类似的问题，关于检测到的封闭的本质表面，使用一个模块理论的角度来看，由Chesebro开发的字符品种。特别地，我们希望构造一个不动点族的双曲3-流形，其环面边界包含闭本质曲面，这些曲面不能被特征簇检测到。我们还将探讨一个封闭的曲面和有界曲面的奇异斜率之间的连接，这是弱检测的字符品种。该奖项是东亚和太平洋夏季研究所计划的一部分，由NSF和澳大利亚科学院共同资助，支持美国研究生的夏季研究。

To understand the structure of a 3-dimensional shape, or 3-manifold, it often helps to study certain surfaces in the 3-manifold that encode important information. Culler-Shalen theory provides the most general tools for constructing such surfaces and has been instrumental in the resolution of many well-known questions. These special surfaces split naturally into two classes: those with boundary and those without. Much of Culler-Shalen theory's power comes from a deep understanding of the surfaces with boundary that one can construct with these tools. On the other hand, very little is known about the associated surfaces without boundary. In this project, we will use a novel algebraic perspective on Culler-Shalen theory to study the associated surfaces without boundary. Our work will resolve numerous problems concerning Culler-Shalen theory itself and will also open the door to applying the theory to an even broader class of questions about 3-manifolds. This research will be conducted in collaboration wth Dr. Stephan Tillman, a leading expert on Culler-Shalen theory and 3-manifolds, at the University of Sydney in Sydney, Australia.More specifically, Culler-Shalen theory uses the algebraic geometry of a 3-manifold's character variety and a construction, due to Stallings, to build essential surfaces in the manifold. The A-polynomial and the Culler-Shalen norm both determine precisely which boundary slopes arise in this fashion when the manifold's boundary consists of a single torus. However, not every boundary slope is detected by Culler-Shalen theory. We will address an analogous question concerning the detected closed essential surfaces using a module-theoretic perspective on character varieties developed by Chesebro. In particular, we hope to construct an in nite family of hyperbolic 3-manifolds with torus boundary containing closed essential surfaces that are not detected by the character variety. We will also explore the connection between the singular slopes of a closed surface and the bounded surfaces which are weakly detected by the character variety. This award, under the East Asia and Pacific Summer Institutes program, supports summer research by a U.S. graduate student and is jointly funded by NSF and the Australian Academy of Science.