William M. Goldman - Topology and dynamics of moduli spaces of geometric structures on surfaces

The space of representations of the fundamental group of a surface into a Lie group is a natural object which exhibits rich geometric structure and rich symmetry. In particular these moduli spaces are interesting examples of symplectic geometry. The ergodicity of the modular group acting on moduli space (for certain compact G) will be discussed, as the relationship to uniformization of geometric structures. When Gis noncompact, we conjecture that there are components where the action is ergodic as well as components where the action is properly discontinuous (arising from uniformizations). Uniformizations by convex domains in the real projective plane, as well as complex hyperbolic Kleinian groups will also be discussed. Using singular hyperbolic structures, one can interpret arbitrary surface group homomorphisms in terms of uniformizations, which provides a geometric interpretation to the dynamical questions about moduli spaces.