Compactification of higher Teichmuller spaces
closed surface of genus . View Teichmuller space as injective and discrete representations of fundamental group to , up to conjugation. Replacing by a larger Lie groups leads to higher Teichmuller theory. This was started by Hitchin 25 years ago.
The part I am interested in today is the compactification of Teichmuller space. This is understood yet only in two cases,
– groups of Hermitian type: a subset of representations, maximal representations, correspond to Teichmuller space; maximal means that a certain numerical invariant, subject to a universal inequality, achieves its maximal possible value;
– groups of real adjoint split type: the Hitchin component plays this role.
The goal is to study ultralimits of maximal (resp. Hitchin) representations. I will focus on the maximal case.
1. Thurston-Parreau compactification
Map a representation to the (projectivized) set of translation lengths of group elements. Translation lengths are measured in the action on the symmetric space of the receiving group .
This can be alternatively described in terms of ultralimits. Fix a non-principal ultrafilter . Fix a finite generating set of . Consider a sequence of representations where some origin in the symmetric space is moved farther and farther away. Pick a sequence of scales such that the maximal displacement of is . Then there is a limiting isometric action on the ultralimit of . It has translation lengths.
Theorem 1 (Burger-Pozzetti 2015)
- is injective.
- splits into a finite disjoint union of subsurfaces along the collection of closed geodesics whose translation length is 0 but such that the translation length of every closed geodesic that intersects is non-zero.
- The restriction of to the fundamental group of (with appropriate choice of base-point in )
- either has a global fixed point,
- or every non-peripheral element has non-zero translation length.
1.2. Robinson fields
We can describe when by merely replacing the reals by a larger real closed field. Start with the ultraproduct . Fix the infinitesimal corresponding to the sequence . Consider for some , and for all . Then the Robinson field is
This is a non-Archimedean field, with an absolute value . Define a distance on positive diagonal matrices as the -norm of logs of eigenvalues. This extends into a pseudo-distance on Siegel’s upper half-space.
Theorem 2 (Parreau) is isometric to the Siegel upper half space associated with modded out by vanishing distances.
1.3. Boundary maps
Maximal representations admit continuous equivariant maps from to the space of triples of Lagrangian subspaces. Using it, one can define a cross-ratio on . This allows to express translation lengths of hyperbolic elements. Say a pair is somewhat short if for every such that is positively oriented, .
Theorem 3 Any somewhat short geodesic is simple.
This is a key tool in the following theorem.
Corollary 4 (Burger-Iozzi-Parreau-Pozzetti) Assume that all hyperbolic elements have positive translation length is the limiting action. Then
- either there exists a syrface filling minimal measured lamination and an equivariant isometric embedding of the corresponding -tree into (note that the action of on this tree has small stabilizers),
- or acts properly discontinuously on and the locally quotient has positive injectivity radius.