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 \ 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 \ 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 \ M -invariant symplectic structure. The main result is that \ 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. . .