Moebius structures on boundaries, III
Today’s material is taken from Jonas Beyrer’s PhD. Given a space, Bourdon’s formula
often takes value 0. Nevertheless, for higher rank symmetric spaces, or for products of spaces, the restriction to the Furstenberg boundary is nontrivial.
1. Symmetric spaces
Write , where is the stabilizer of some point . Let be the Killing orthogonal complement of is , let be a maximal abelian subalgebra. It generate flat totally geodesic subspaces in . It is split into convex polyhedral sectors called Weyl chambers. Each of them contributes a spherical simplex to the visual boundary (Weyl chambers at infinity). be the group of parabolic isometries which fixes pointwise a Weyl chamber
The Furstenberg boundary is the set of Weyl chambers at infinity. Since , where .
Weyl chambers at infinity have a well-defined center, a point of the Weyl chamber at infinity. So one can view as a subset of the visual boundary.
Our main examples are and .
In a product of spaces, flats are products of geodesics from factors. Weyl chambers are quadrants, Weyl chambers at infinity are fourth of circles. The Furstenberg boundary is the product of ideal boundaries of factors. Two pairs and are opposite iff and .
consists of trace free diagonal matrices. A maximal flat corresponds to a set of quadratic forms (or ellipsoids) diagonal in the same basis. At infinity, a Weyl chamber is a segment ending with quadratic forms with matrices and respectively. The center point corresponds to .
2.1. Opposite pairs
Say two Weyl chambers and at infinity are opposite if they are at maximal combinatorial distance. This is the generic case. If so, there is a geodesic joining their center points and . One defines
This varies in , and is equal to 1 iff belongs to the geodesic . For other pairs of Weyl chambers at infinity, . does not satisfy triangle inequality.
Definition 1 A 4-tuple of Furstenberg boundary points is admissible if at most one pair consists of non-opposite points.
The cross-ratio triple of an admissible 4-tuple is
Claim. The cross-ratio does not depend on the choice of the origin .
Note this cross-ratio is not Ptolemaic.
Two pairs and are opposite iff and . Here is the formula for , .
is Heisenberg group. The Weyl chambers at infinity which are opposite to the previously described base Weyl chamber at infinity is an orbit of . If
then the distance to of the image of is
It vanishes for , showing that indeed, generic pairs are opposite.
3. Modified cross-ratios
What if one changes center points for other points of Weyl chambers at infinity? Each Weyl chamber at infinity has a unique developing map to the model Weyl chambers at infinity. Points in opposite Weyl chambers at infinity developped to the same point are opposite, i.e. joined by a geodesic.
The formula changes to
where is the chosen developing point in the model Weyl chamber at infinity.
It seems that Labourie uses some of these cross-ratios to characterize Anosov representations: representations inducing the same cross-ratio are conjugate.
With Buyalo, we are trying to describe the set of Moebius structures on the circle.