** Geometric structures, compactifications of representation varieties, and non archimedean geometry, III **

**1. To infinity (and beyond) **

** 1.1. Compactification **

We intend to describe the boundary of the representation variety .

Theorem 1 (Burger-Pozzetti)If is a maximal representation, then

- Nontrivial elements do not have eigenvalues of modulus 1. In particular, translation lengths are .
- If and represent intersecting closed geodesics in , then the following analogue of Collar Lemma holds,

In particular, the vector-valued length function never vanishes.

Corollary 2 (Parreau)The projectived length map

has relatively compact image and any boundary point is the vector-valued length function of a -action on a building associated to where is a Robinson field.

There is a loss of information.

**Question 1**. Does this building have special geometric properties?

**Question 2**. Is there a way to organize all these actions on buildings into a coherent compactification of ?

** 1.2. Answer to question 1 **

The following is known since the 1980’s.

Theorem 3 (Skora)If , boundary points are exactly length functions of actions on -trees with small stabilizers, i.e. stabilizers of germs of segments are either trivial or cyclic.

For , we need study sequences in and the resulting actions an asymptotic cones of .

A sequence leaves to infinity if some marked point, say , is moved farther and farther away by some element of a fixed generating system . Pick a sequence such that

Pick a non-principal ultrafilter . Form the corresponding asumptotic cone . It is a complete metric space, with an isometric action of .

**Example**. . Assume pinches some closed geodesic , and does not affect curves disjoint from . Due to the Collar Lemma, any closed curve intersecting has length tending to infinity at speed length. In the limit, has a fixed point.

Definition 4A simple closed geodesic on isspecialif

- whenever represents .
- For any closed geodesic intersecting , whenever represents .

There are at most special geodesics.

Theorem 5The isometric action is faithful. Let be a component of the complement in of special geodesics. There is a dichotomy:

- Either (PT): every curve in which is not a boundary component has .
- Or (FP): fixes a point in .

This is used in the proof of

Theorem 6 (Burger-Pozzetti-Iozzi-Parreau)The action on is small.

An other tool is the theory of maximal representations of surfaces with boundary. Indeed, the scale is too large for certain components, thus a scale needs be chosen for each component. Only finitely many scales arise.

Personnally, I am not too fluent with buildings, I prefer the language of Robinson fields.