Geometric structures, compactifications of representation varieties, and non archimedean geometry, III
1. To infinity (and beyond)
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 4 A simple closed geodesic on is special if
- whenever represents .
- For any closed geodesic intersecting , whenever represents .
There are at most special geodesics.
Theorem 5 The 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.