Geometry and dynamics of representations into semisimple Lie Groups

PI 计划建立关于有限生成群的表示种类的特殊子集到半简单李群的几何和动力学性质的新结果。除了微分几何、刚性理论和代数群的经典方法外，连续有界上同调的新技术也将得到重要的应用。该项目建立在之前与 M. Burger 和 A. Iozzi 的合作基础上，其中使用有界 Kaehler 类（第二个连续有界上同调类）来建立任意有限生成群到 Hermitian 型李群的 Zariski 稠密表示的刚性结果，以及定义和研究曲面群表示变体的连通分量，从而给出 Teichmueller 空间的推广。该项目的一部分涉及研究这些特殊组件中表面群表示的精细几何特性、类似于 Fenchel-Nielsen 的参数化或 Teichmueller 空间上的剪切坐标，以及这些广义 Teichmueller 空间与希格斯丛的模空间和表面上的局部齐次几何结构之间的显式关系。在该项目的第二部分中，有界凯勒类研究中使用的方法和技术将扩展到更高程度的定义和研究有界上同调类，这有望产生新的刚性现象和关于其他半单李群表示的结构结果。对称性在自然界中随处可见，在生物学、化学、物理和数学中非常重要。从数学上来说，研究对称性的一种方法是考虑对称群，即保留给定对称性的变换群。具有多种对称性的图案通常是最佳配置，因此稳定或刚性，因为小的变形或变化会破坏对称性。有时，具有对称性的图案会出现在族中并形成所谓的“模空间”；那么，小的甚至大的变形即使改变了图案也不会破坏对称性。在这个项目中，这两种现象都是通过对物体几何复杂性的特殊测量来检测和研究的，这仅取决于它们的对称性。

The PI plans to establish new results concerning geometric and dynamical properties of special subsets of the representation variety of a finitely generated group into a semisimple Lie group. Besides classical methods from differential geometry, rigidity theory and algebraic groups, new techniques from continuous bounded cohomology will be employed in an essential way. The project builds upon previous joint work with M. Burger and A. Iozzi,in which the bounded Kaehler class, a second continuous bounded cohomology class, was used, both, to establish rigidity results for Zariski dense representations of arbitrary finitely generated groups into Lie groups of Hermitian type, as well as to define and study connected components of the representation variety of a surface group, which give generalizations of Teichmueller space. One part of the project concerns the study of refined geometric properties of surface group representations in these special components, of parametrizations similar to Fenchel-Nielsen or shear coordinates on Teichmueller space, and of the explicit relations between these generalized Teichmueller spaces and the moduli spaces of Higgs bundles and of locally homogeneous geometric structures on the surface. In a second part of the project, methods and techniques used in the study of the bounded Kaehler class will be extended to define and investigate bounded cohomology classes in higher degree, which are expected to give rise to new rigidity phenomena and structural results about representations into other semisimple Lie groups.Symmetries arise everywhere in nature and are very important in biology, chemistry, physics and mathematics. Mathematically one way to study symmetries is to consider symmetry groups, that is groups of transformations preserving the given symmetries. Patterns with many symmetries are often optimal configurations and as such stable or rigid, because small deformations or changes destroy the symmetry. Sometimes patterns with symmetries arise in families and form so called "moduli spaces"; then, small or even large deformations do not break symmetry even though they change the pattern. In this project both phenomena are detected and studied using a special measurement for the geometric complexity of objects, which depends only on their symmetries.