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.
