Notes of Viktor Schroeder’s Cambridge lecture 11-01-2017

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.

1. 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.
2. Consider the subset of points ${P({\mathbb R}_+^3)}$ of projective plane. It is a triangle. To a quadruple of distinct points, associate

$\displaystyle \begin{array}{rcl} crt=(d_{12}d_{34},d_{13}d_{24},d_{14}d_{23}) \end{array}$

This map is equivariant under ${\mathfrak{S}_4}$.

3. Buyalo prefers to take -logarithm of distances

A submöbius structure is a triple of maps ${\mathbb{X}_A,\mathbb{X}_B,\mathbb{X}_C}$ from distinct quadruples to ${(0,\infty)}$.

semi-metric submöbius structures satisfy the following cocycle identity

$\displaystyle \begin{array}{rcl} \mathbb{X}_C(\alpha,x,y,\beta)\mathbb{X}_C(\alpha,y,z,\beta)=\mathbb{X}_C(\alpha,x,z,\beta)/ \end{array}$

Theorem 1 (Buyalo) A submöbius structure arises from a semi-metric iff it satisfies the cocycle identity.

Indeed, define a semi-metric by

$\displaystyle \begin{array}{rcl} d_{\alpha,\beta,\omega}(x,y)=\frac{\mathbb{X}_A(\alpha,x,\omega,\beta)\mathbb{X}_A(\alpha,\omega,y,\beta)}{\mathbb{X}_A(\alpha,x,y,\beta)}. \end{array}$

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. ${CAT(-1)}$

In a ${CAT(-1)}$ space ${X}$, fix an origin ${o}$ and define a metric on the ideal boundary

$\displaystyle \begin{array}{rcl} d_o(x,y)=e^{-(x|y)_o}. \end{array}$

It is indeed a metric (Bourdon). Changing origin ${o}$ leads to a Möbius equivalent distance. Therefore ${\partial X}$ 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 ${\alpha}$ and ${\beta}$, say that ${x,y}$ belong to the same sphere relative to ${\alpha}$ and ${\beta}$ 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 ${\sigma}$ such that for every quadruple ${q}$ on ${\sigma}$, ${crt(q)}$ belongs to the boundary of the triangle with vertices ${(0:1:1)}$, ${(1:0:1)}$ and ${(1:1:0)}$. 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 ${\partial {\mathbb C} H^n}$, there are ${{\mathbb R}}$-circles and ${{\mathbb C}}$-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 ${CAT(0)}$ spaces ?