Notes of Viktor Schroeder’s first informal Cambridge lecture 16-05-2017

Moebius structures on boundaries, I

This is an informal series of 3 lectures. I start with boundaries of hyperbolic groups. I will continue with Furstenberg boundaries of higher rank symmetric spaces (joint work with my student Beyrer).

1. Moebius structures

4 distinct points in a set can be split into pairs in 3 ways, whence an epimorphism ${\mathfrak{S}_4\rightarrow\mathfrak{S}_3}$, with kernel

$\displaystyle \begin{array}{rcl} \{1,(12)(34),(13)(24),(14)(23)\}. \end{array}$

Say a quadruple is regular if all points are distinct, and admissible if no 3 of them coincide.

1.1. Semi-metrics

A semi-metric on a set is a function ${X\times X\rightarrow[0,+\infty]}$ which is

• symmetric,
• positive on distinct pairs,
• at most one point can have infinite distance to an other point. For this point, distance to all points is infinity.

1.2. Cross-ratios

The Moebius structure of a semi-metric can be defined in 3 equivalent ways. Given a quadruple ${(x_1,x_2,x_3,x_4)\in X}$, let ${\rho_{ij}=\rho(x_i,x_j)}$ and define cross-ratio

$\displaystyle \begin{array}{rcl} \frac{\rho_{12}\rho_{34}}{\rho_{14}\rho_{23}}. \end{array}$

The resulting 6 numbers (after permutations) can be organized in different ways.

1. View

$\displaystyle \begin{array}{rcl} crt_\rho:Reg_4\rightarrow \Sigma=\{(a:b:c)\in{\mathbb R} P^2\,;\,a,b,c>0\}. \end{array}$

This extends to ${Adm_4\rightarrow \hat\Sigma=\Sigma}$ union 3 points. Under permutation, ${crt_\rho}$ changes via ${\mathfrak{S}_3}$ as above.

2. View

$\displaystyle \begin{array}{rcl} \mathbb{X}:Reg_4\times\Theta\rightarrow{\mathbb R}_+, \end{array}$

where ${\Theta}$ is the 3-point set of splittings in pairs. The product of the 3 functions equals 1. Under permutation, ${\mathbb{X}}$ changes via taking inverse and ${\mathfrak{S}_3}$ acting on ${\Theta}$.

3. Alternatively, one may replace values by their logarithms.

1.3. Sub-Moebius structures

Definition 1 A sub-Moebius structure on a set ${X}$ is a function ${crt:Ad_4\rightarrow\hat\Sigma}$ satisfying

1. Normalization. ${crt(x,x,y,z)=(0:1:1)}$.
2. Symmetry. ${crt(\pi(q))=\phi(\pi(crt(q)))}$.

1.4. The cocycle condition

Not all sub-Moebius structures arise from semi-metrics. Those arising from semi-metrics satisfy an extra equation, the cocycle condition:

$\displaystyle \begin{array}{rcl} crt(\alpha,x,y,\beta)crt(\alpha,y,z,\beta)=crt(\alpha,x,z,\beta). \end{array}$

Theorem 2 (Buyalo) A sub-Moebius structure arises from a semi-metric if and only if it satisfies the cocycle condition.

Indeed, set

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

This is a semi-metric. Different choices of ${\alpha,\beta,\omega}$ define the same sub-Moebius structure.

1.5. Moebius equivalent semi-metrics

Say two semi-metrics are Moebius equivalent if they define the same sub-Moebius structure. Here are constructions of semi-metrics Moebius equivalent to a given one ${\rho}$.

• Multiplication with a constant ${\lambda}$. ${\lambda\rho}$.
• Involution. Given ${\omega\in X}$,

$\displaystyle \begin{array}{rcl} \rho_\omega(x,y)=\frac{\rho(x,y)}{\rho(x,\omega)\rho(\omega,y)}. \end{array}$

• Multiplication with a positive function ${\lambda}$.

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

2. Boundaries

Let ${X}$ be a ${CAT(-1)}$ metric space. Fix origin ${o\in X}$. Then (Bourdon) the semi-metric

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

is a metric. Changing ${o}$ gives a Moebius equivalent metric.

2.1. The Ptolemaic inequality

Moebius structures on boundaries of ${CAT(-1)}$ spaces satisfy an extra inequality, which we call 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.

${CAT(0)}$ metrics are Ptolemaic. The sphere with the chordal metric (i.e. isometric to a subset of Euclidean space) is Ptolemaic. The sphere in its Riemannian metric is not Ptolemaic.

Thus triangle inequality does not imply Ptolemaic. Conversely, Ptolemaic inequality does not imply triangle inequality for all metrics in the class. However, if a sub-metric Ptolemaic Moebius structure has a point at infinity, then it satisfies triangle inequality. Also the Moebius class contains bounded sub-metrics which are metrics.

2.2. Hyperbolic groups

Mineyev has constructed metrics on hyperbolic groups whose visual distances define a Ptolemaic Moebius structure. More on this next time.