The space of representations of the fundamental group of a surface into a Lie group is a natural object with rich geometry and symmetry. The topology of the surface influences the algebraic structure of the
deformation space, with natural families of Hamiltonian flows related to curves on the surfaces. In particular these flows define continuous deformations closely related to the action of the discrete
mapping class group of the surface. In the case of the one-holed torus and the four-holed sphere, these deformation spaces are families of affine cubic surfaces with actions of the modular group
by polynomial Poisson automorphisms.