Towards higher Teichmuller theory, IV
1. To infinity and beyond
Today, . We consider maximal representations . They are quasi-isometric embeddings. The class is deformation invariant. If , they coincide with holonomies of hyperbolic structures on .
The goal is to understand diverging sequences of maximal representations, from different points of view.
2. A simplified version of Parreau-Thurston compactification
Simplified means that I gave up explaining the Weyl chamber valued length function.
acts on Siegel upper half-space
The translation length of an element is given by the formula
where are the eigenvalues of . Note that the characteristic polynomial of a symplectic matrix is palindromic, whence the sum of terms only. Define as the map , .
Corollary 1 (Parreau 2010) The map
has relatively compact image . Any belonging to its closure isthe length function of a -isometric action on a building associated to , where is an appropriate Robinson field.
This is not the end of the story.
Question 1. Do the limiting actions on buildings have special geometric properties?
Question 2. Can one organize the actions on buildings into a compactification of ?
For instance, if , Skora’s theorem asserts that the closure is precsely the set of actions with small edge stabilizers.
Definition 2 An isometric action of on a space has small stabilizers if the stabilizer of a germ of segment is trivial or cyclic.
2.1. Results on question 1
Fix a non-principal ultrafilter on . Let be a sequence of representations. Fix a base point in . Say a sequence of scales is adapted if for some finite generating set ,
Then one sets . Let
This is a space, thus translation lengths are well defined. There is a limiting action of .
Question. How are the elements with vanishing translation lengths look like?
It is true that .
Theorem 3 (Pozzetti 2015) Fix a hyperbolic metric on (for convenience). Say a closed geodesic is special if
- For all that cross , .
Then special curves split into subsurfaces with geodesic boundary .
Theorem 4 (Burger-Pozzetti 2015) The limiting representation is faithfull. For every , the following dichotomy holds:
- either (PT): For every non-peripheral , .
- or (FP): The restriction of to has a global fixed point.
To go further, one needs treat surfaces with boundary separately. Note that maximality makes sense. Adapted scales for each can be chosen. The process ends because has finite complexity.
Theorem 5 (Burger-Iozzi-Parreau-Pozzetti 2017) The limiting representation has small stabilizers.
The proof depends on the answer to Question 2.
2.2. Answer to Question 2
We follow Brumfiel’s real spectrum compactification of real algebraic spaces. Points there are homomorphisms of the ring of functions to real closed fields.
Point stabilizers, if not cyclic, contain surface groups and the restriction of the representation is again maximal. They act on the link of the fixed point. We want to view this as an action on a spherical building, associated to a certain residue field. Brumfiel tells us that this action belongs to the compactification, hence is a limit of maximal representations.
2.3. Proof of Theorem 5
Maximality passes to the limit (maximality is defined in terms of existence of a transversality preserving boundary map between Grassmannian Lagrangians; this is a difficult transversality result).
Germs of arc stabilizers, if not cyclic, contain surface groups and the restriction of the representation is again maximal. Since maximal representations do not fall into parabolic subgroups, such a subgroup cannot fix a direction, contradiction.