Notes of Marc Burger’s third Leverhulme lecture 24-04-2017

Towards higher Teichmuller theory, III

1. On integer points in the Hitchin moduli space

Let ${G}$ be a simple Lie group of real split type (${Sp(n,{\mathbb R}),Sl(n,{\mathbb R}),SO(n,n+1)}$…). Then ${Hom(\Gamma_g,G)}$ has a Hitchin component. In the sequel, we focus on ${G=PSl(n,{\mathbb R})}$. The Hitchin component is the connected component which contains the Fuchsian representations

$\displaystyle \begin{array}{rcl} \mathcal{F}_n(S_g)&=&\{\pi_n\circ\rho_h\,;\,\rho_h:\Gamma_g\rightarrow PSl(2,{\mathbb R})\textrm{ orientation preserving holonomy} \\ &&\textrm{ representation of a hyperbolic structure }h\textrm{ on }S_g\}. \end{array}$

where ${\pi_n:PSl(2,{\mathbb R})\rightarrow PSl(n,{\mathbb R})}$ is the irreducible representation.

We denote by ${Hit_n(S_g)}$ its image in

$\displaystyle \begin{array}{rcl} Rep(\Gamma_g,PSl(n,{\mathbb R}))=Hom(\Gamma_g,PSl(n,{\mathbb R}))^{ss}/PSl(n,{\mathbb R}). \end{array}$

Examples.

1. If ${n=2}$, ${Hit_2(S_g)}$ has only Fuchsian representations, it is Teichmüller space, of dimension ${6g-6}$.
2. If ${n=3}$, ${Hit_3(S_g)}$ is the space of marked convex projective sutructures on ${S_g}$ (convex means that the universal convering space is a convex open set in ${{\mathbb R} P^2}$.

Facts.

1. ${Hit_n(S_g)}$ has dimension ${(2g-2)\mathrm{dim}(Sl(n,{\mathbb R}))}$ (Hitchin 1992).
2. Mapping class group ${Mod_g}$ acts properly discontinuously on it (Labourie 2006).
3. Every Hitchin representation has an open domain of discontinuity in

1.1. Classical ideas

Many features of Teichmüller theory generalize:

The Weil-Petersson metrics generalizes into the Pressure metric (Bridgemann-Canary-Labourie-Sambarino).

The Bonahon-Thurston shear coordinates can be generlized too (Bonahon-Dreyer).

Thurston-Penner coordinates have been generalized by Fock-Gontcharov.

The Collar Lemma has been generalized by Zhang-Lee.

1.2. New features

However, new phenomena show up:

Entropy rigidity behaves differently (Potrie-Sambarino).

Integer points become of interest.

2. Integral representations

Definition 1 Let ${\rho:\Gamma_g\rightarrow Sl(n,{\mathbb R})}$ be a representation.

1. Say ${\rho}$ is integral if all traces are integers.
2. Given a lattice ${\Lambda be a lattice. Say ${\rho}$ is ${\Lambda}$-integral if ${\rho(\Gamma_g)<\Lambda}$.

Let ${Hit_n^{\mathbb Z}(S_g)}$, ${Hit_n^\Gamma(S_g)}$ denote the sets of such representations.

These are ${Mod_g}$-invariant subsets.

Question. Is ${Mod_g\setminus Hit_n^\Gamma(S_g)}$ finite ? Is ${Mod_g\setminus Hit_n^{\mathbb Z}(S_g)}$ finite ?

2.1. Dimension 2

This turns out to be the case for ${n=2}$. Nevertheless, the two cases are inequivalent. The proof of the second statement is much harder.

Example. Let ${a,b}$ be square-free integers. Let ${H^{a,b}}$ denote the corresponding quaternion algebra (i.e. the 4-dimensional ${{\mathbb Q}}$-algebra with basis ${1,i,j,k}$ and relations ${i^2=a,j^2=b,ij=-ji=k}$). Then unit elements form a group isomorphic to ${Sl(2,{\mathbb R})}$, integer units form a surface group. Its genus is given by a complicated formula. All these examples, and all their finite index subgroups, give points of

$\displaystyle \begin{array}{rcl} \bigcup_{g>1}Hit_2^\mathbb Z(S_g). \end{array}$

However, modulo ${Mod_g}$, there are only finitely many points with given genus. This is by no means obvious from the genus formula.

2.2. Dimension 3

Theorem 2 (Long-Reid-Thislethwaite) If ${n=3}$, ${Mod_g\setminus Hit_3^{\mathbb Z}(S_g)}$ is infinite.

This is a by-product of their rational parametrization of ${Hit_3}$. Applied to a 1-parameter family of representations of a ${(3,3,4)}$ triangle group, it produces Zariski-dense subgroups which are pairwise non commensurable. The open convex sets on which they act are projectively inequivalent, they converge to a triangle. This has to do with the fact that there is a limiting action on a triangle building.

2.3. Higher dimensions

The sequel is ongoing work with Francois Labourie and Anna Wienhard.

Proposition 3 For all ${n\geq 4}$, ${Mod_g\setminus Hit_n^{\mathbb Z}(S_g)}$ is infinite.

This does not follow from the 3-dimensional case. Indeed, a 3D representation, viewed as a 4D representation, is not Hitchin.

Since this set is infinite, we want to measure its growth in terms of some notion of height.

Recall that the translation length of ${g\in Sl(n,{\mathbb R})}$ when acting on the symmetric space is equal to the Euclidean norm of the vector of ${\log}$ of absolute values of eigenvalues of matrix ${g}$. Given two representations ${\rho}$ and ${\pi}$, set

$\displaystyle \begin{array}{rcl} A(\rho,\pi)=\sup_{\gamma\in\Gamma_g}\frac{\ell(\rho(\gamma))}{\ell(\pi(\gamma))}. \end{array}$

For ${n=2}$, this is called Thurston’s asymmetric distance.

Fact. If ${\pi}$ is Fuchsian, ${\ell(\pi(\gamma))}$ is bounded below by the conjugacy length of ${\gamma}$ (min of length if conjugates with respect to a word metric).

Here is our notion of height: it measures distance to Fuchsian representations.

Definition 4

$\displaystyle \begin{array}{rcl} h(\rho)=\inf_{\pi\in\mathcal{F}_n(S_g)}A(\pi,\rho). \end{array}$

Then ${h}$ is a ${Mod_g}$-invariant function.

Theorem 5 (Burger-Labourie-Wienhard) The number of Hitchin representations of height ${\leq T}$, mod the mapping class group, is finite.

The minimal area of a representation is the infimal area of an equivariant map to the symmetric space. We can show that the number of Hitchin representations of minimal area ${\leq T}$, mod the mapping class group, is finite. This is clear for ${\Lambda}$-integral representations, since then equivariant maps descend to maps to a fixed compact of finite volume manifold.