Moebius structures on boundaries, II
1. Ptolemaic Moebius structures
Recall that a Moebius structure is Ptolemaic if cross-ratios satisfy the Ptolemaic inequality,
This means that takes its values in the triangle with vertices at the extra points .
- Boundaries of -spaces, in their Bourdon metric, are Ptolemaic.
- Boundaries of hyperbolic groups admit natural Ptolemaic Moebius structures. This follows from a construction by Mineyev-Yu, as observed by Nica. Furthermore, Nica-Spakula observed that visual metrics associated to Green metrics associated to random walks also define Prolemaic Moebius structures.
There is an indirect link between Ptolemaic and triangle inequalities.
- Given a bounded Ptolemaic Moebius structure, pick a point and form the semi-metric which sends at infinity. Then is a metric.
- Two metrics in the Moebius class which send the same point to infinity are proportional.
- A Ptolemaic Moebius structure always contains at least one bounded metric. Indeed, given any bounded semi-metric in the structure, add an ideal point et extend by , where is an arbitrary origin in . Then is a bounded metric in the Moebius structure of .
2. The space of metrics of a Moebius structure
Given two Moebius-equivalent metrics and , there exists a Lipschitz function such that
One denotes by
Question. Is every Moebius space the boundary of some space in a natural sense ?
This space should be the space of metrics in the Moebius structure. The map
can be viewed as a Busemann function on . It is well defined up to an additive constant, since at each point,
Kingshook Biswas defines as the subset of diameter 1, antipodal metrics in the Moebius structure. Antipodal means that every point has an other point lying at distance 1 from it. Then Biswas embeds into by . The image is a closed subset, inherits a proper metric (balls are compact) that does not depend on the choice of reference metric . He shows that, in the case of the boundary of a space , isometrically embeds in , which is within bounded Hausdorff distance from .
3. Hausdorff measure
Assume that is the Haudorff dimension of . Then Hausdorff measures corresponding to different metrics in the Moebius structures differ by a factor
Therefore the measure on pairs
is invariant on .
Cross-ratio define a kind of distance on pairs.
4.1. Spheres between points
Given distinct points , the sets form a 1-parameter family of “spheres” separating from . Indeed, by the cocycle condition, this is an equivalence relation.
4.2. Jorgensen’s inequality
I learned this from J. Parker and S. Markham. Given a loxodromic transformation , with axis , set
Then are cross-ratios. Define
Proposition 1 Let be a compact Moebius space. let be a discrete group of Moebius transformations of . Then for every loxodromic elements , and
5. Next time
I will explain Beyrer’s Moebius structure on the Furstenberg boundary of higher rank symmetric spaces.