CAREER: Kleinian, Arithmetic and Mapping Class Groups

在这个项目中，PI将研究三种特殊的群，即Kleinian群，映射类群和高阶李群中的格。这些群体尽管有很大的不同，但却显示出惊人的相似之处。这可以用相关几何对象（如局部对称空间、模空间等）的刚性特性来解释。该项目的目标是研究这些相关几何对象的刚性如何反映在问题群的代数性质中。更具体地说，PI将研究Kleinian群的生成数与相关双曲3流形的几何观测值之间的关系，例如拉普拉斯的体积或谱。为了做到这一点，有必要发展克莱因群变形理论的一些迄今尚未探索的方面。PI还将研究马古利斯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间调和映射的研究来解决。来自动力学的工具也将被应用于研究映射类群的某些动作的刚性。最后，PI将研究局部对称空间的最小棘的存在性及其几何、同调和代数性质。在这种情况下，主要的工具将是代数拓扑和代数群理论中的经典论证的结合。群是一个具有根本意义的代数对象。也许最有趣的群体出现在一些几何结构的对称性上，比如一个晶体、一个物理模型、一个空间物体等等。例如，Kleinian群在分形对象的研究中起着重要的作用。算术群是矩阵的群，因此在数学的所有分支中都有应用。最后，映射类群，曲面的对称群，不仅在数学中起着中心作用，而且在理论物理中也很重要。已知这三类群都具有有用的刚性性质，但尚不清楚这种刚性如何与群的代数性质有效地互锁。获得有效的，至少在原则上，可计算的刚度结果是这个建议的一个组成部分。除了提高数学这一重要领域的知识水平外，培养研究生独立研究的能力也是该项目的主要目标之一。PI打算应用的广泛的工具和方法将极大地促进这一点。这种广度也将有助于在数学界传播已获得的结果。

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.