Towards higher Teichmuller theory, III
1. On integer points in the Hitchin moduli space
Let be a simple Lie group of real split type (…). Then has a Hitchin component. In the sequel, we focus on . The Hitchin component is the connected component which contains the Fuchsian representations
where is the irreducible representation.
We denote by its image in
- If , has only Fuchsian representations, it is Teichmüller space, of dimension .
- If , is the space of marked convex projective sutructures on (convex means that the universal convering space is a convex open set in .
- has dimension (Hitchin 1992).
- Mapping class group acts properly discontinuously on it (Labourie 2006).
- 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 be a representation.
- Say is integral if all traces are integers.
- Given a lattice be a lattice. Say is -integral if .
Let , denote the sets of such representations.
These are -invariant subsets.
Question. Is finite ? Is finite ?
2.1. Dimension 2
This turns out to be the case for . Nevertheless, the two cases are inequivalent. The proof of the second statement is much harder.
Example. Let be square-free integers. Let denote the corresponding quaternion algebra (i.e. the 4-dimensional -algebra with basis and relations ). Then unit elements form a group isomorphic to , 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
However, modulo , 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 , is infinite.
This is a by-product of their rational parametrization of . Applied to a 1-parameter family of representations of a 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 , 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 when acting on the symmetric space is equal to the Euclidean norm of the vector of of absolute values of eigenvalues of matrix . Given two representations and , set
For , this is called Thurston’s asymmetric distance.
Fact. If is Fuchsian, is bounded below by the conjugacy length of (min of length if conjugates with respect to a word metric).
Here is our notion of height: it measures distance to Fuchsian representations.
Then is a -invariant function.
Theorem 5 (Burger-Labourie-Wienhard) The number of Hitchin representations of height , 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 , mod the mapping class group, is finite. This is clear for -integral representations, since then equivariant maps descend to maps to a fixed compact of finite volume manifold.