## 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.