Notes of Viktor Schroeder’s third informal Cambridge lecture 30-05-2017

Moebius structures on boundaries, III

Today’s material is taken from Jonas Beyrer’s PhD. Given a {CAT(0)} space, Bourdon’s formula

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

often takes value 0. Nevertheless, for higher rank symmetric spaces, or for products of {CAT(-1)} spaces, the restriction to the Furstenberg boundary is nontrivial.

1. Symmetric spaces

Write {M=G/K}, where {K} is the stabilizer of some point {o}. Let {\mathfrak{p}} be the Killing orthogonal complement of {\mathfrak{k}} is {\mathfrak{g}}, let {\mathfrak{a}\subset\mathfrak{p}} be a maximal abelian subalgebra. It generate flat totally geodesic subspaces in {M}. It is split into convex polyhedral sectors {\mathfrak{a}^+} called Weyl chambers. Each of them contributes a spherical simplex to the visual boundary (Weyl chambers at infinity). {N^+} be the group of parabolic isometries which fixes pointwise a Weyl chamber

The Furstenberg boundary {\partial_F M} is the set of Weyl chambers at infinity. Since {G=KAN^+}, {\partial_F M=G/P} where {P=Z_K(A)AN^+}.

Weyl chambers at infinity {c} have a well-defined center, a point {x_c} of the Weyl chamber at infinity. So one can view {\partial_F M} as a subset of the visual boundary.

Our main examples are {H^2\times H^2} and {Sl(3,{\mathbb R})/SO(3)}.

1.1. Products

In a product of {CAT(-1)} spaces, flats are products of geodesics from factors. Weyl chambers are quadrants, Weyl chambers at infinity are fourth of circles. The Furstenberg boundary is the product of ideal boundaries of factors. Two pairs {(x_1,x_2)} and {(y_1,y_2)} are opposite iff {x_1\not=x_2} and {y_1\not=y_2}.

1.2. {Sl(3,{\mathbb R})/SO(3)}

{\mathfrak{a}} consists of trace free diagonal matrices. A maximal flat corresponds to a set of quadratic forms (or ellipsoids) diagonal in the same basis. At infinity, a Weyl chamber is a segment ending with quadratic forms with matrices {diag(1,1,-2)} and {diag(2,-1,-1)} respectively. The center point corresponds to {diag(1,0,-1)}.

2. Cross-ratio

2.1. Opposite pairs

Say two Weyl chambers {x} and {y} at infinity are opposite if they are at maximal combinatorial distance. This is the generic case. If so, there is a geodesic joining their center points {x_c} and {y_c}. One defines

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

This varies in {(0,1]}, and is equal to 1 iff {o} belongs to the geodesic {(x_c,y_c)}. For other pairs of Weyl chambers at infinity, {\rho_o=0}. {\rho_o} does not satisfy triangle inequality.

Definition 1 A 4-tuple of Furstenberg boundary points is admissible if at most one pair consists of non-opposite points.

The cross-ratio triple of an admissible 4-tuple is

\displaystyle  \begin{array}{rcl}  (\rho_o(x_1,x_2)\rho_o(x_3,x_4):\rho_o(x_1,x_3)\rho_o(x_2,x_4),\rho_o(x_1,x_4)\rho_o(x_2,x_3)). \end{array}

Claim. The cross-ratio does not depend on the choice of the origin {o}.

Note this cross-ratio is not Ptolemaic.

2.2. Products

Two pairs {(x_1,x_2)} and {(y_1,y_2)} are opposite iff {x_1\not=x_2} and {y_1\not=y_2}. Here is the formula for {\rho_o}, {o=(o_1,o_2)}.

\displaystyle  \begin{array}{rcl}  \rho_o((x_1,x_2)(y_1,y_2))=\rho_{o_1}(x_1,x_2)^{\mu}\rho_{o_2}(y_1,y_2))^\mu, \end{array}

where {\mu=1/\sqrt{2}}.

2.3. {Sl(3,{\mathbb R})/SO(3)}

{N^+} is Heisenberg group. The Weyl chambers at infinity which are opposite to the previously described base Weyl chamber at infinity {p} is an orbit of {N^+}. If

\displaystyle  \begin{array}{rcl}  N(\alpha,\beta,\gamma)=\begin{pmatrix} 1&\alpha &\gamma+\frac{1}{2}\alpha\beta \\ 0 & 1 & \beta\\ 0&0& 1 \end{pmatrix}, \end{array}

then the distance to {p} of the image of {p} is

\displaystyle  \begin{array}{rcl}  \sqrt{|\gamma^2-\frac{1}{4}\alpha^2\beta^2} \end{array}

It vanishes for {\gamma=\pm\frac{1}{2}\alpha\beta}, showing that indeed, generic pairs are opposite.

3. Modified cross-ratios

What if one changes center points for other points of Weyl chambers at infinity? Each Weyl chamber at infinity has a unique developing map to the model Weyl chambers at infinity. Points in opposite Weyl chambers at infinity developped to the same point are opposite, i.e. joined by a geodesic.

3.1. Products

The formula changes to

\displaystyle  \begin{array}{rcl}  \rho_o((x_1,x_2)(y_1,y_2))=\rho_{o_1}(x_1,x_2)^{\mu_1}\rho_{o_2}(y_1,y_2))^{\mu_2}, \end{array}

where {(\mu_1,\mu_2)} is the chosen developing point in the model Weyl chamber at infinity.

3.2. {Sl(3,{\mathbb R})/SO(3)}

It seems that Labourie uses some of these cross-ratios to characterize Anosov representations: representations inducing the same cross-ratio are conjugate.

With Buyalo, we are trying to describe the set of Moebius structures on the circle.


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
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