** 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.

**1. Cross-ratios **

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.

**2. **

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 ?