EAPSI: Surface Subgroups in Gromov-Thurston Manifolds and Brownian Motion in Riemannian Manifolds of Negative Curvature

EAPSI：格罗莫夫-瑟斯顿流形中的表面子群和负曲率黎曼流形中的布朗运动

• 批准号: 1614366
• 负责人: Mehrzad Monzavi
• 金额: $ 0.54万
• 依托单位: Monzavi Mehrzad
• 依托单位国家: 美国
• 项目类别: Fellowship Award
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-15 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1614366&HistoricalAwards=false
• 关键词: EAPSI; Surface; Subgroups; Gromov; Thurston

Geometric group theory comes into view from the fact that abstract mathematical objects such as groups can be viewed as geometric objects and studied with geometric tools. Geometric group theory reformulates problems from different areas of mathematics in a geometric framework. Surface groups played a great role in the resolution of many long-standing conjectures during the last few years. The tools used in geometric group theory lead to applications in diverse areas of mathematics including topology, geometry and ergodic theory. In this project, the surface subgroups of a certain manifold (a topological space) denoted by Gromov-Thurston manifold will be constructed. It will lead to a better understanding of properties of more general spaces such as compact manifolds of negative curvature. Furthermore, some dynamical properties of Riemannian manifolds of negative curvature will be explored. This research will be conducted in collaboration with Dr. Seonhee Lim, a noted expert on geometric group theory and dynamics, at Seoul National University in Seoul, South Korea.There are two specific goals to this project. The first goal concerns Gromov's famous question about surface subgroups. "Does every one-ended word-hyperbolic group contain a surface subgroup isomorphic to the fundamental group of a closed surface of genus at least two?" In the case of the fundamental groups of hyperbolic 3-manifolds, this question is the famous Surface Subgroup Theorem. It yielded to the proof of Virtual Haken Conjecture, Virtual Fibered Conjecture and Ehrenpreis Conjecture. This project will investigate the construction of surface subgroups in the Fundamental group of a Gromov-Thurston manifold. It would lead to establishment of surface subgroups in more general cases such as compact manifolds of negative curvature. The second goal of this project is to study the properties of random walks on reductive groups in Riemannian manifolds of negative curvature.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 National Research Foundation of Korea.

几何群论的产生是由于抽象的数学对象，如群，可以被看作几何对象，并可以用几何工具来研究。几何群论将不同数学领域的问题在几何框架中重新表述。在过去几年中，地表群在解决许多长期存在的猜想方面发挥了巨大作用。在几何群论中使用的工具导致应用在不同的数学领域，包括拓扑，几何和遍历理论。本课题将构造以Gromov-Thurston流形表示的某流形（拓扑空间）的曲面子群。它将使我们更好地理解更一般空间的性质，例如负曲率的紧致流形。此外，还探讨了负曲率黎曼流形的一些动力学性质。这项研究将与Seonhee Lim博士合作进行，他是韩国首尔国立大学几何群论和动力学方面的知名专家。这个项目有两个具体目标。第一个目标与Gromov关于表面子群的著名问题有关。“是否每个单端词双曲群都包含一个与至少两个属的封闭曲面的基本群同构的曲面子群？”对于双曲3流形的基本群，这个问题就是著名的曲面子群定理。给出了虚哈肯猜想、虚纤维猜想和Ehrenpreis猜想的证明。本课题将研究Gromov-Thurston流形基本群中曲面子群的构造。这将导致在诸如负曲率紧致流形等更一般情况下曲面子群的建立。本课题的第二个目标是研究负曲率黎曼流形中约化群上的随机漫步的性质。该奖项是由美国国家科学基金会（NSF）和韩国国立科学研究财团共同资助的东亚太平洋暑期研究所项目，旨在支持美国研究生的暑期研究。