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

几何群论的出现是因为抽象的数学对象，如群，可以被看作几何对象，并可以用几何工具来研究。几何群论在几何框架中重新表述了不同数学领域的问题。在过去的几年里，表面基团在许多长期存在的问题的解决中发挥了重要作用。在几何群论中使用的工具导致在不同数学领域的应用，包括拓扑学，几何学和遍历理论。在这个项目中，我们将构造一个流形（拓扑空间）的曲面子群，这个流形用Gromov-Thurston流形表示。这将导致更好地理解更一般的空间，如负曲率的紧致流形的性质。此外，还将讨论负曲率黎曼流形的一些动力学性质。这项研究将与韩国首尔的首尔国立大学的著名几何群论和动力学专家Seonhee Lim博士合作进行。第一个目标涉及格罗莫夫著名的问题曲面子群。“是否每个单端字双曲群包含一个曲面子群同构于亏格至少为2的闭曲面的基本群？在双曲三维流形的基本群的情况下，这个问题是著名的曲面子群定理。虚Haken猜想、虚纤维猜想和Escherapreis猜想的证明。本计画将研究Gromov-Thurston流形的基本群中曲面子群的构造。这将导致建立表面子群在更一般的情况下，如紧流形的负曲率。该项目的第二个目标是研究负曲率黎曼流形中约化群上的随机游动的性质。该奖项属于东亚和太平洋夏季研究所计划，支持美国研究生的夏季研究，由NSF和韩国国家研究基金会共同资助。

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.