** 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 .

** 1.1. Ultralimits **

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 3Any 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.