The dynamics of classifying geometric structures
1. Marked geometric structures
Moduli spaces of geometric structures do not all behave like the moduli space of Riemann surfaces: in general, it is not a well behaved space, it is a quotient by a group action with interesting dynamics.
Lie and Klein (1872), Ehresmann (1936) suggest to study -structures on manifolds . Experience shows that it is useful to introduce a deformation space of marked -structures, on which the mapping class group acts. A marking is the data of a -manifold and a diffeomorphism .
In some cases (e.g. hyperbolic structures on surfaces), this action is properly discontinuous, resulting in a quotient space which is a manifold mere singularities. In general, it is not.
1.1. Example: complete affine surfaces
All Euclidean structures on the 2-torus are affinelu isomorphic. Other affine structures, discovered by Kuiper, are obtained from the polynomial diffeomorphism
Indeed, change of charts turn out to be affine.
The mapping class group acts ergodically on the deformation space (Moore 1966).
2. Moduli spaces of representations
Let be a closed surface, . Let be a simple Lie group. Connected components of are indexed by .
With Forni, we try to use Teichmuller dynamics, and replace the difficult action by a simpler action. This is defined on the unit tangent bundle of Teichmuller space .
Let . This is a bundle over . Let be its unit tangent bundle.
Theorem 1 (Forni-Goldman) For compact, the Teichmuller flow is strongly mixing on .
Each element of defines a character function, hence a Hamiltonian flow. Dehn twists suffice to generate the ring of functions, hence
2.1. An example: compact surfaces of Euler characteristic
There are 4 of them, all have . was determined as early as 1889. It is isomorphic (as a complex manifold) to .
The function is invariant under (Nielsen). Level sets have invariant symplectic structures. Interesting involutions arise as deck transformations of branched double coverings given by coordinate projections to .
Level sets for values in contain a component corresponding to unitary representations, on which the action is ergodic.
The case of the once-punctured Klein bottle is particularly interesting. The action does not extend to projective space.