Ergodic Theory on Moduli Spaces

. Let M be a compact surface with Ø(M ) ! 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to SU(2) (for example SU(2)). Then the mapping class group \Γ M of M acts on the moduli space X(M ) of flat G-bundles over M (possibly twisted by a fixed central element of G). When M is closed, then \Γ M preserves a symplectic structure on X(M ) which has finite total volume on M . More generally, the subspace of X(M ) corresponding to flat bundles with fixed behavior over @M carries a \Γ M -invariant symplectic structure. The main result is that \Γ M acts ergodically on X(M ) with respect to the measure induced by the symplectic structure. Contents 1. Introduction 2 1.1. Statement of results 2 1.2. The Narasimhan-Seshadri foliation of the universal moduli space 3 1.3. Moduli spaces over surfaces with boundary 4 1.4. Parabolic structures 5 1.5. Further speculations 5 1.6. Outline of proof 7 1.7. Acknowledgements 10 2. Preliminarie...