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

几何群论产生于这样一个事实，即抽象的数学对象，如群，可以被视为几何对象，并利用几何工具进行研究。几何群论在几何框架内对不同数学领域的问题进行了重新表述。在过去的几年里，表面群在解决许多长期存在的猜测中发挥了巨大的作用。几何群论中使用的工具导致了数学的各个领域的应用，包括拓扑学、几何学和遍历理论。在这个项目中，我们将构造一个由Gromov-瑟斯顿流形表示的流形(一个拓扑空间)的曲面子群。它将使我们更好地理解更一般的空间的性质，例如负曲率的紧致流形。此外，还将探讨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国立大学的著名几何群论和动力学专家林善熙博士合作进行。这个项目有两个具体的目标。第一个目标与格罗莫夫关于表面子群的著名问题有关。“是否每个单端字双曲群都有一个曲面子群同构于亏格至少为2的闭曲面的基本群？”对于双曲3-流形的基本群，这个问题就是著名的表面子群定理。它屈服于虚Haken猜想、虚Fibered猜想和Ehrenpreis猜想的证明。本课题将研究格罗莫夫-瑟斯顿流形的基本群中的曲面子群的构造。这将导致在更一般的情况下建立曲面子群，例如负曲率的紧致流形。这个项目的第二个目标是研究负曲率黎曼流形上约化群上的随机游动的性质。这个奖项由东亚和太平洋夏季学院计划资助一名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.