Abstract

The moduli space of complex curves of genus g is known to be the quotient of the Teichmüller space of complex curves by the action of the group of isotopy classes of diffeomorphisms of a surface of genus g (the mapping class group).

The fact that this action be as nice as to make of the quotient (the moduli space of curves) a well-behaved topological space (an orbifold) has always struck me as a kind of a miracle. 

Instead of looking at parameter spaces of complex structures on surfaces, one can look at all sorts of different geometric structures (projective, hyperbolic, affine, lorentzian, representations of surface groups, you-name-it). 

I will discuss the question of the well-behavedness of the action of the mapping class group in those cases, and try to highlight the role played by dynamical systems that one can attach to geometric structures on surfaces.