Geometry and dynamics of representations into semisimple Lie Groups

PI计划建立关于有限生成群到半单李群的表示簇的特殊子集的几何和动力学性质的新结果。除了来自微分几何、刚性理论和代数群的经典方法外，还将以一种必要的方式使用来自连续有界上同调的新技术。该项目建立在与M.Burger和A.Iozzi以前合作的基础上，其中使用了有界Kaehler类，第二个连续的有界上同调类，两者都建立了任意有限生成群的Zariski稠密表示为Hermitian型李群的刚性结果，并定义和研究了表面群表示簇的连通分支，这给出了Teichmueller空间的推广。该项目的一个部分是研究这些特殊分支中表面群表示的精细几何性质，研究Teichmueller空间上类似于Fichel-Nielsen坐标或剪切坐标的参数化，以及这些广义Teichmueller空间与Higgs丛的模空间和曲面上的局部齐次几何结构之间的显式关系。在项目的第二部分，研究有界Kaehler类的方法和技巧将被扩展到定义和研究更高阶的有界上同调类，这有望产生关于表示为其他半单李群的新的刚性现象和结构结果。对称性在自然界中无处不在，在生物、化学、物理和数学中都是非常重要的。从数学上讲，研究对称的一种方法是考虑对称群，即保持给定对称的变换群。具有许多对称性的图案通常是最优的配置，因此是稳定的或刚性的，因为微小的变形或变化破坏了对称性。有时具有对称性的图案以族的形式出现并形成所谓的“模空间”；然后，小的甚至大的变形不会破坏对称性，即使它们改变了图案。在这个项目中，这两种现象都是使用一种特殊的测量对象几何复杂性的方法来检测和研究的，这种度量只取决于对象的对称性。

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.