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 ?

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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