Notes of Marc Burger’s first Leverhulme lecture 15-03-2017

Towards higher Teichmuller theory

Study homomorphisms of surface groups to real algebraic groups {G\subset Gl(n,{\mathbb R})}. A surface group {\Gamma} is the fundamental group of a closed surface {S} of genus {g} (boundaries will show up only later). Since {\Gamma} has a {2g}-generators 1-relator presentation, {Hom(\Gamma,G)} is defined by 1 matrix equation in {G^{2g}}. In particular, it is a real algebraic subset of {Gl(n,{\mathbb R})^{2g}}.

Aim: identify connected components of {Hom(\Gamma,G)} formed by representations with geometric signficance.

This depends on {G}. When {G} is real split (i.e. {Sl(n,{\mathbb R})}, {Sp(2n,{\mathbb R})}, {SO(n,n+1)},…), there is a preferred component known as Hitchin component. When {G} is of Hermitian type (e.g. {Sp(2n,{\mathbb R})}, {SO(2,n)}, {SU(p,q)},…), there is a notion of maximal representation.

1. Case of {Sl(2,{\mathbb R})}

Holonomies of hyperbolic structures on surfaces form an open subset of {Hom(\Gamma,PSl(2,{\mathbb R}))}. One shows that the induced map on Teichmuller space

\displaystyle  \begin{array}{rcl}  \mathcal{T}=Diff^+_0(S)\setminus Hyp(S)\rightarrow Hom(\Gamma,PSl(2,{\mathbb R}))/PSl(2,{\mathbb R}) \end{array}

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 {S} into pairs of pants. A marked hyperbolic structure on {S} determines a length and an angle for each cuff, showing that {\mathcal{T}} is homeomorphic to {{\mathbb R}^{6g-6}}. In particular, it is connected, thus its image is a component of {Hom(\Gamma,PSl(2,{\mathbb R}))/PSl(2,{\mathbb R})}. 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 {\rho}, which may have a dense image, more care is needed in the definition. Let {V_\rho} denote the hyperbolic plane bundle over {S} associated to {\rho} (quotient of {\tilde S\times H^2} by diagonal action of {\Gamma}). Since {H^2} is contractible, there exist a smooth section. Pull-back the area form (defined on {H^2} fibers, it extends by horizontality to a 2-form on {V_\rho}) and integrate it, this gives the area invariant {T(\rho)}.

This number coincides with the Euler number of the associated circle bundle (replace {\tilde S} with {\partial\tilde S}); therefore it is an integer.

Milnor inequality (1958):

\displaystyle  \begin{array}{rcl}  |T(\rho)|\leq 2g-2. \end{array}

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

Theorem 1 (Goldman 1980) {\rho} is the holonomy of an orientation compatible hyperbolic structure iff {T(\rho)=2g-2}.

1.3. Dynamical characterization

Fix a background hyperbolic metric on {S}. Let {g_t} denote its geodesic flow. It lifts to a flow {\tilde \rho_t} on {E_\rho=\Gamma\setminus(T_1 \tilde S\times {\mathbb R}^2)}.

When {\rho} is the holonomy of a hyperbolic structure, there exists an equivariant homeomorphism {\tilde S\rightarrow H^2} which extends to an equivariant map {\phi:\partial\tilde S\rightarrow\partial H^2=P^1({\mathbb R})}. The trajectory of a point of {\partial S} has two endpoints in {P^1({\mathbb R})} which we interpret as lines in {{\mathbb R}^2}, or rather, in fibers of {E_\rho}.

We obtain a splitting {E_\rho=E_\rho^+\oplus E_\rho^-} into two continuous sub-bundles (no additional smoothness) which are {\tilde g_t}-invariant, and have exponential contraction-dilation properties.

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

Liouville measure on {T_1 S} is ergodic.

2. Hitchin component

Let {\pi_n:PSl(2,{\mathbb R})\rightarrow PSl(n,{\mathbb R})} be the irreducible representation. Let {\mathcal{F}_n} be the set of representations of the form {\pi_n\circ\rho_h}, where {\rho_h} is the holonomy of a hyperbolic structure.

Definition 3 The Hitchin component {H_n(S)=Hom_H(\Gamma,PSl(n,{\mathbb R}))/PSl(n,{\mathbb R})} where the set {Hom_H(\Gamma,PSl(n,{\mathbb R}))} of Hitchin representations is the connected component of the image of {\mathcal{F}_n} in {Hom(\Gamma,PSl(n,{\mathbb R})}.

The starting point of the whole theory is the following

Theorem 4 (Hitchin) {H_n(S)} is homeomorphic to {{\mathbb R}^d}, {d=|\chi(S)|\mathrm{dim}(Sl(n,{\mathbb R}))}.

What is its geometric significance ? A hint is provided by the 3-dimensional case. Say that a projective structure on {S} is convex if is covered by a convex subset of {P^2({\mathbb R})}.

Theorem 5 (Choi-Goldman 1997) The space of marked convex projective structures on {S} up to isotopy is homeomorphic to {H_3(S)}.

For higher values of {n}, 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.

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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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