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

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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.