** Towards higher Teichmuller theory **

Study homomorphisms of surface groups to real algebraic groups . A surface group is the fundamental group of a closed surface of genus (boundaries will show up only later). Since has a -generators 1-relator presentation, is defined by 1 matrix equation in . In particular, it is a real algebraic subset of .

**Aim**: identify connected components of formed by representations with geometric signficance.

This depends on . When is real split (i.e. , , ,…), there is a preferred component known as Hitchin component. When is of Hermitian type (e.g. , , ,…), there is a notion of maximal representation.

**1. Case of **

Holonomies of hyperbolic structures on surfaces form an open subset of . One shows that the induced map on Teichmuller space

is injective. Hence Teichmuller theory is indeed the study of a subset of the representation variety.

For other real algebraic groups, it is not so clear what should replace Teichmuller space.

** 1.1. Fenchel-Nielsen coordinates **

Cut into pairs of pants. A marked hyperbolic structure on determines a length and an angle for each cuff, showing that is homeomorphic to . In particular, it is connected, thus its image is a component of . General principles imply that it is defined by finitely many equations and inequations. Such an explicit description is available for small values of genus only.

** 1.2. Area of a representation **

For a hyperbolic structure, it is the area. For more general representations , which may have a dense image, more care is needed in the definition. Let denote the hyperbolic plane bundle over associated to (quotient of by diagonal action of ). Since is contractible, there exist a smooth section. Pull-back the area form (defined on fibers, it extends by horizontality to a 2-form on ) and integrate it, this gives the area invariant .

This number coincides with the Euler number of the associated circle bundle (replace with ); therefore it is an integer.

**Milnor inequality (1958)**:

Due to Gauss-Bonnet theorem, equality holds for hyperbolic structures.

Theorem 1 (Goldman 1980)is the holonomy of an orientation compatible hyperbolic structure iff .

** 1.3. Dynamical characterization **

Fix a background hyperbolic metric on . Let denote its geodesic flow. It lifts to a flow on .

When is the holonomy of a hyperbolic structure, there exists an equivariant homeomorphism which extends to an equivariant map . The trajectory of a point of has two endpoints in which we interpret as lines in , or rather, in fibers of .

We obtain a splitting into two continuous sub-bundles (no additional smoothness) which are -invariant, and have exponential contraction-dilation properties.

Theorem 2is the holonomy of a hyperbolic structure if and only if splits into two continuous -invariant rank 1 sub-bundles which have exponential contraction-dilation properties.

Liouville measure on is ergodic.

**2. Hitchin component **

Let be the irreducible representation. Let be the set of representations of the form , where is the holonomy of a hyperbolic structure.

Definition 3The Hitchin component where the set of Hitchin representations is the connected component of the image of in .

The starting point of the whole theory is the following

Theorem 4 (Hitchin)is homeomorphic to , .

What is its geometric significance ? A hint is provided by the 3-dimensional case. Say that a projective structure on is *convex* if is covered by a convex subset of .

Theorem 5 (Choi-Goldman 1997)The space of marked convex projective structures on up to isotopy is homeomorphic to .

For higher values of , one does not have such a clear-cut interpretation. A substitute has been provided by Labourie, this will be the subject of the next lecture.