## Notes of Viktor Schroeder’s second informal Cambridge lecture 23-05-2017

Moebius structures on boundaries, II

1. Ptolemaic Moebius structures

Recall that a Moebius structure is Ptolemaic if cross-ratios satisfy the Ptolemaic inequality,

$\displaystyle \begin{array}{rcl} \rho_{12}\rho_{34}\leq \rho_{23}\rho_{14}+\rho_{13}\rho_{24}. \end{array}$

This means that ${crt}$ takes its values in the triangle ${\Delta\subset\hat\Sigma}$ with vertices at the extra points ${(1:1:0),(1:0:1),(0:1:1)}$.

Examples

1. Boundaries of ${CAT(-1)}$-spaces, in their Bourdon metric, are Ptolemaic.
2. 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 ${\omega\in X}$ and form the semi-metric ${\rho_\omega}$ which sends ${\omega}$ at infinity. Then ${\rho_\omega}$ 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 ${\hat X=X\cup\{\hat\omega\}}$ et extend ${\rho}$ by ${\hat\rho(x,\hat\omega)=\rho(x,o)+1}$, where ${o}$ is an arbitrary origin in ${X}$. Then ${\hat\rho_{\hat\omega}}$ is a bounded metric in the Moebius structure of ${X}$.

2. The space of metrics of a Moebius structure

Given two Moebius-equivalent metrics ${\rho}$ and ${\tau}$, there exists a Lipschitz function ${\lambda}$ such that

$\displaystyle \begin{array}{rcl} \rho(x,y)=\lambda(x)\lambda(y)\tau(x,y). \end{array}$

One denotes by

$\displaystyle \begin{array}{rcl} \lambda=(\frac{d\rho}{d\tau})^{1/2}. \end{array}$

Question. Is every Moebius space the boundary of some space in a natural sense ?

This space should be the space ${\mathcal{M}}$ of metrics in the Moebius structure. The map

$\displaystyle \begin{array}{rcl} \rho\mapsto -\log\frac{d\rho}{d\tau}(x) \end{array}$

can be viewed as a Busemann function on ${\mathcal{M}}$. It is well defined up to an additive constant, since at each point,

$\displaystyle \begin{array}{rcl} \frac{d\rho}{d\tau}\frac{d\tau}{d\sigma}=\frac{d\rho}{d\sigma}. \end{array}$

Kingshook Biswas defines ${\mathcal{M}_a}$ 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 ${\mathcal{M}_a}$ into ${C^0(X)}$ by ${\rho\mapsto -\log\frac{d\rho}{d\tau}}$. The image is a closed subset, ${\mathcal{M}_a}$ inherits a proper metric (balls are compact) that does not depend on the choice of reference metric ${\tau}$. He shows that, in the case of the boundary of a ${CAT(-1)}$ space ${Y}$, ${Y}$ isometrically embeds in ${\mathcal{M}_a}$, which is within bounded Hausdorff distance from ${Y}$.

3. Hausdorff measure

Assume that ${D}$ is the Haudorff dimension of ${(X,\rho)}$. Then Hausdorff measures corresponding to different metrics in the Moebius structures differ by a factor

$\displaystyle \begin{array}{rcl} \frac{d\mu_\rho}{d\mu_\tau}(x)=(\frac{d\rho}{d\tau}(x))^D. \end{array}$

Therefore the measure on pairs

$\displaystyle \begin{array}{rcl} d\nu(x,y)=\rho(y,s)^{-2D}d\mu_\rho(x)d\mu_\rho(y) \end{array}$

is invariant on ${X\times X}$.

Cross-ratio define a kind of distance on pairs.

4. Methods

4.1. Spheres between points

Given distinct points ${p,q,y\in X}$, the sets ${\{x\in X\,;\,[x:y:p:q]=1\}}$ form a 1-parameter family of “spheres” separating ${p}$ from ${q}$. 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 ${\alpha\in Moeb(X)}$, with axis ${p,q}$, set

$\displaystyle \begin{array}{rcl} a= \rho(p,q)\rho(z,\alpha z),\quad b=\rho(z,p)\rho(\alpha z,q),\quad c=\rho(\alpha z,p)\rho(z,q). \end{array}$

Then ${\frac{a}{b},\frac{b}{c},\frac{c}{a}}$ are cross-ratios. Define

$\displaystyle \begin{array}{rcl} m_\alpha=\sup_z \frac{a}{b}\frac{a}{c}. \end{array}$

Proposition 1 Let ${X}$ be a compact Moebius space. let ${\Gamma}$ be a discrete group of Moebius transformations of ${X}$. Then for every loxodromic elements ${\alpha,\beta\in\Gamma}$, ${d_\alpha}$ and ${d_\beta}$

$\displaystyle \begin{array}{rcl} m_\alpha^2(d((p,q),(\beta p,\beta q))+1)\geq 1. \end{array}$

5. Next time

I will explain Beyrer’s Moebius structure on the Furstenberg boundary of higher rank symmetric spaces.