三维流形基本群的极限群

In his monumental work on the elementary theory of free groups, Sela undertook a deep study of the logic of free groups, and answered an old question of Tarski. In this work, a surprising and deep connection between geometric group theory and logic is established. Limit groups is one of the key objects studied by Sela in this work. Due to the importance of the Tarski problem and the geometric flavor of the proof, limit groups has attracted geometric group theorists' attention and has inspired a large amount of activity in geometric group theory. In this project, we will generalize the concept of Sela's limit group and define limit groups over 3-manifold groups. We will investigate whether this new type of limit groups share similar properties with Sela's limit groups. In particular, we will try to prove that any sequence of epimorphisms between limit groups over 3-manifold groups contains finitely many proper epimorphisms. As an application, we hope to answer a question about epimorphisms between 3-manifold groups proposed by Alan Reid, Shicheng Wang and Qing Zhou.

Sela开创性地运用了几何群论的工具解决了逻辑中一个著名问题——Tarski问题。这项深刻的工作建立了几何群论和逻辑模型论之间令人惊讶的联系。 极限群是Sela在这项工作中深入研究的核心对象之一。对极限群及其拓展的研究一直是几何群论中的重要方向。 本项目将继续拓展极限群的概念并将其与三维流形的基本群结合，从而定义三维流形群极限群。我们将探索三维流形极限群是否与极限群拥有类似的性质。其中同态满射序列有限性将被重点研究。作为应用， 我们希望能回答一个由Alan Reid， 王诗宬和周青提出的关于三维流形群的群同态满射序列的问题。