CAREER: Kleinian, Arithmetic and Mapping Class Groups

在这个项目中，PI将研究三类特殊的群，即Klein群、映射类群和高阶Lie群中的格。尽管这些群体截然不同，但它们表现出惊人的相似之处。这可以用局部对称空间、模空间等相关几何对象的刚性性质来解释。这个项目的目标是研究这些相关几何对象的刚性如何反映在所讨论的群的代数性质中。更具体地说，PI将研究Klein群的生成元数目与相关的双曲3-流形的几何观察值之间的关系，例如拉普拉斯的体积或谱。为了做到这一点，有必要发展克雷尼亚群形变理论的一些迄今未被探索的方面。PI还将研究Marguis超刚性定理在多大程度上适用于映射类群之间的同态。这个问题将通过组合方法和模空间之间的调和映射的研究来解决。动力学中的工具也将被应用于研究映射类组的某些动作的刚性。最后，PI将研究局部对称空间的极小脊线的存在性及其几何、同调和代数性质。在这种情况下，主要的工具将是代数拓扑学和代数群论中的经典论点的组合。群是一个基本感兴趣的代数对象。也许最有趣的群出现在某些几何结构的对称性上，例如晶体、物理模型、太空中的物体等。例如，Kleian群在研究分形物中起着重要的作用。算术组是矩阵的组，因此在数学的所有分支中都有应用。最后，映射类群，曲面的对称性群，不仅在数学中起着核心作用，而且在理论物理等方面也很重要。众所周知，这三类群都具有有用的刚性性质，但这种刚性如何有效地与群的代数性质相关联尚不清楚。获得有效的、至少在原则上可计算的刚性结果是这项建议的一个组成部分。除了提高一个重要的数学领域的知识状况外，培养研究生进行独立研究是这个项目的主要目标之一。国际和平研究所打算采用的各种工具和方法将极大地促进这一点。这种广度还将促进数学界对已有结果的反驳。

In this project the PI will study three particular classes of groups, namely Kleinian groups, mapping class groups and lattices in higher rank Lie groups. In spite of being quite different, these groups show remarkable similarities. This may be explained by the rigidity properties of associated geometric objects such as locally symmetric spaces, moduli space,... The goal of this project is to study how the rigidity of these associated geometric objects is reflected in the algebraic properties of the groups in question. More concretely, the PI will study the relations between the number of generators of a Kleinian group and geometric observables of the associated hyperbolic 3-manifold, such as the volume or the spectrum of the Laplacian. In order to do so, it will be necessary to develop some so-far unexplored aspects of the deformation theory of Kleinian groups. The PI will also study to what extent the Margulis superrigidity theorem holds for homomorphisms between mapping class groups. This problem will be approached via combinatorial methods and also via the study of harmonic maps between moduli spaces. Tools from dynamics will also be applied to study the rigidity of certain actions of mapping class groups. Finally, the PI will study the existence of minimal spines for locally symmetric spaces and their geometric, homological and algebraic properties. In this case, the main tools are going to be a combination of classical arguments in algebraic topology and in the theory of algebraic groups.A group is an algebraic object of fundamental interest. Perhaps the most interesting groups appear as symmetries of some geometric structure, such as a crystal, a physical model, an object in space, etc... For instance, Kleinian groups play an important role in the study of fractal objects. Arithmetic groups are groups of matrices and hence have applications in all branches of mathematics. Finally, the mapping class group, the group of symmetries of a surface, plays not only a central role in mathematics, but is also important in for instance theoretical physics. All three classes of groups are known to have useful rigidity properties but it is unknown how this rigidity is effectively interlocked with the algebraic properties of the groups. It is an integral part of this proposal to obtain effective and, at least in principle, computable rigidity results. Besides advancing the state of knowledge in an important area of mathematics, educating graduate students to do independent research is one of the main goals of this project. This will be greatly facilitated by the broad spectrum of tools and methods that the PI intends to apply. This breadth will also facilitate the disemination of the obtained results among the mathematical community.