Möbius geometry of a semi-metric
Joint work with Sergei Buyalo and my students T. Förtsch, J. Beyrev.
Semi-metric means that we do not require triangle inequality, and that infinite values are admitted.
Here are 3 equivalent ways to organize cross-ratios.
- 4 distinct points in a (semi-)metric space define 6 cross-ratios, indexed by the 3 partitions of 4 letters in two pairs (plus their inverses). Well chosen, their product equals 1.
- Consider the subset of points of projective plane. It is a triangle. To a quadruple of distinct points, associate
This map is equivariant under .
- Buyalo prefers to take -logarithm of distances
A submöbius structure is a triple of maps from distinct quadruples to .
semi-metric submöbius structures satisfy the following cocycle identity
Theorem 1 (Buyalo) A submöbius structure arises from a semi-metric iff it satisfies the cocycle identity.
Indeed, define a semi-metric by
Note that several semi-metrics define the same submöbius structure.
If the resulting semi-metric is a metric (resp. ultra metric), we speak of a Möbius (resp. ultrametric Möbius) structure.
Completeness, or doubling, are Möbius invariant properties.
In a space , fix an origin and define a metric on the ideal boundary
It is indeed a metric (Bourdon). Changing origin leads to a Möbius equivalent distance. Therefore has a canonical Möbius structure, which is Ptolemaic. Ptolemaic means that the Ptolemaic inequality holds.
A modification of this construction extends to hyperbolic groups. The invariant Möbius structure is Ptolemaic (Buyalo, Mineyev).
3. Concepts of Möbius geometry
Given two distinct points and , say that belong to the same sphere relative to and if the cross ratio equals 1. This is an equivalence relation (due to the cocycle identity).
A circle in a Möbius space is a topological circle such that for every quadruple on , belongs to the boundary of the triangle with vertices , and . This is the Ptolemaic identity, equality case in the Ptolemaic inequality.
Among Möbius spaces, ideal boundaries of rank 1 symmetric spaces should play a prominent role. They can be characterized by their wealth in circles. For instance, for , there are -circles and -circles also known as chains.
Ideal boundaries of geodesically complete trees have ultrametric Möbius structures. Conversely, a complete ultrametric Möbius bounds a geodesically complete tree.
4. Furstenberg boundaries of higher rank symmetric spaces
A Gromov product can be defined on it, by passing to the limit. Cross-ratios arise, leading to a Möbius structure whose Möbius morphisms coincide with isometries of the symmetric case.
Is there some Möbius structure on some quotient of the visual boundary of certain spaces ?