## Notes of Bill Goldman’s Cambridge lecture 23-06-2017

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 ${(G,X)}$-structures on manifolds ${S}$. Experience shows that it is useful to introduce a deformation space of marked ${(G,X)}$-structures, on which the mapping class group ${\pi_0(Diff(S))}$ acts. A marking is the data of a ${(G,X)}$-manifold ${S'}$ and a diffeomorphism ${S\rightarrow S'}$.

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

$\displaystyle \begin{array}{rcl} (x,y)\mapsto (x+y^2,y). \end{array}$

Indeed, change of charts turn out to be affine.

The mapping class group ${Gl(2,{\mathbb Z})}$ acts ergodically on the deformation space (Moore 1966).

2. Moduli spaces of representations

Let ${S}$ be a closed surface, ${\pi=\pi_1(S)}$. Let ${G}$ be a simple Lie group. Connected components of ${Rep(\pi,G)}$ are indexed by ${\tau\in\pi_1([G,G])}$.

With Forni, we try to use Teichmuller dynamics, and replace the difficult ${MCG}$ action by a simpler ${{\mathbb R}}$ action. This is defined on the unit tangent bundle of Teichmuller space ${T(S)}$.

Let ${E=(T(S)\times Rep(\pi,G)_\tau)/MCG}$. This is a bundle over . Let ${U}$ be its unit tangent bundle.

Theorem 1 (Forni-Goldman) For ${G}$ compact, the Teichmuller flow is strongly mixing on ${U}$.

Each element of ${\pi}$ 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 ${-1}$

There are 4 of them, all have ${\pi_1=F_2}$. ${Rep(\pi,Sl(2))/Sl(2)}$ was determined as early as 1889. It is isomorphic (as a complex manifold) to ${{\mathbb C}^3}$.

The function ${k=Tr([\rho(X),\rho(Y)])}$ is invariant under ${Out(F_2)}$ (Nielsen). Level sets have invariant symplectic structures. Interesting involutions arise as deck transformations of branched double coverings given by coordinate projections to ${{\mathbb C}^2}$.

Level sets for values in ${(-2,2)}$ contain a component corresponding to unitary representations, on which the ${Out(F_2)}$ action is ergodic.

The case of the once-punctured Klein bottle is particularly interesting. The ${Out(F_2)}$ action does not extend to projective space.