The space of representations of the fundamental group of a surface S into a Lie group G exhibits rich geometric structure and symmetry. The mapping class group D of S acts on the moduli space of equivalence classes of representations with an invariant symplectic geometry. Furthermore these spaces support natural Poisson algebras of functions and a rich symplectic geometry generalizing both the Weil-Petersson geometry of Teichmuller space and the Kahler form on moduli spaces of vector bundles. When G is compact the D-action is ergodic. For noncompact semisimple Lie groups the moduli space contains the Teichmuller space of S where the action is properly discontinuous.
Interesting dynamics accompanies interesting dynamics.
For surfaces of small genus, we explicitly describe the mapping class action as automorphisms of affine cubic surfaces. Singular uniformizations provide a general geometric framework for understanding general representations of fundamental groups.