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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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