Research in Geometric Group Theory

摘要：相对双曲群是目前几何群论中一个非常活跃的研究领域。它们包括双曲群、几何有限双曲流形的基本群、自由积和极限群（也称为“完全残馀自由群”），因此与组合和几何群论、双曲几何、拓扑学和逻辑直接相关。这个项目的目的是研究相对双曲群的一些逻辑方面。群的初等理论由在群中为真的一阶句组成。这是一个有趣而强大的群体不变量，对它的研究仍处于早期阶段。这个项目将回答关于相对双曲群的初等理论的问题，以及关于与相对双曲群具有相同初等理论的群可以说些什么。这将导致逻辑和几何群论的应用。离散群是数学中许多几何对象的对称集合。由于这个原因，离散群的研究影响着数学的许多领域，同时也受到这些领域的影响。这个项目中考虑的群是相对双曲的群，它们自然地产生于拓扑和双曲几何、组合和几何群论以及群的逻辑研究中。相对双曲群在“负弯曲群”（称为双曲群）和“非正弯曲群”之间形成了一座桥梁，“负弯曲群”已经被很好地理解，而“非正弯曲群”已经被研究过，但在很多方面仍然是一个谜。本课题的目的是从逻辑的角度研究相对双曲群。特别是，形式逻辑句子能告诉我们关于一个群体的什么？只考虑形式逻辑句子是对我们自己的一个相当严格的限制，并导致一个群的自然不变量，称为群的初等理论。初等理论是在一组中为真的形式（一阶）逻辑句子的集合。它是一个强大的不变量，然而有不同的群体具有相同的基本理论。本课题将研究相对双曲群的初等理论，并研究那些与相对双曲群具有相同初等理论的群。这些方法部分来自于逻辑学，同时也来自于几何群论和低维拓扑学。

Abstract:Relatively hyperbolic groups are currently an extremely active area of research in geometric group theory. They encompass hyperbolic groups, the fundamental groups of geometrically finite hyperbolic manifolds, free products and limit groups (also known as `fully residually free groups'), and thus are directly related to combinatorial and geometric group theory, to hyperbolic geometry, to topology and to logic. The purpose of this project is to examine some of the logical aspects of relatively hyperbolic groups. The elementary theory of a group consists of those first order sentences which are true in the group. This is an interesting and powerful invariant of a group, the study of which is still at an early stage. This project will answer questions about the elementary theory of relatively hyperbolic groups, and also what can be said about a group with the same elementary theory as a relatively hyperbolic group. This will lead to applications both to logic and also to geometric group theory.Discrete groups arise as sets of symmetries of many geometric objects in mathematics. For this reason, the study of discrete groups impacts on, and is impacted by, a wide variety of fields in mathematics. The groups under consideration in this project are relatively hyperbolic groups, which arise naturally from within topology and hyperbolic geometry, as well as in combinatorial and geometric group theory and also in the study of the logic of groups. Relatively hyperbolic groups form a bridge between `negatively curved groups' (called hyperbolic groups), which are well understood, and `non-positively curved groups', which have been much studied but in many ways remain a mystery. The purpose of this project is to study relatively hyperbolic groups from the point of view of logic. In particular, what can formal logical sentences tell us about a group? To consider only formal logical sentences is quite a restrictive limit to place upon ourselves, and leads to a natural invariant of a group, called the elementary theory of a group. The elementary theory is the set of formal (first order) logical sentences which are true in a group. It is a powerful invariant, and yet there are different groups with the same elementary theory. This project will study the elementary theory of relatively hyperbolic groups and also investigate those groups with the same elementary theory as a relatively hyperbolic group. The methods come partially from logic but also from geometric group theory and low-dimensional topology.