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<Sl(n,{\mathbb R})} 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.

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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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