The $\mathrm{PSL}(2,\mathbb{R}{)}^2$-configuration space of four points in the torus $S^1\times S^1$

Abstract

The torus $\mathbb{T}=S^1\times S^1$ appears as the ideal boundary ${\partial }_{\infty }Ad{S}^3$ of the three-dimensional anti-de Sitter space $Ad{S}^3$, as well as the Fürstenberg boundary $\mathcal{F}(X)$ of the rank-2 symmetric space $X={\mathrm{SO}}_0(2,2)/\mathrm{SO}(2)\times \mathrm{SO}(2)$. We introduce cross-ratios on the torus in order to parametrise the $\mathrm{PSL}(2,\mathbb{R}{)}^2$ configuration space of quadruples of pairwise distinct points in $\mathbb{T}$ and define a natural Möbius structure on $\mathbb{T}$ and therefore on $\mathcal{F}(X)$ and ${\partial }_{\infty }Ad{S}^3$ as well.

16 pages, 1 table, 1 figure

References in corpus (2)

• A Geometric transition from hyperbolic to anti de Sitter geometry
• Moebius and sub-Moebius structures