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The torus $\mathbb{T}=S^1\times S^1$ appears as the ideal boundary$\partial_\infty AdS^3$ of the three-dimensional anti-de Sitter space $AdS^3$,as well as the F\"urstenberg boundary $\mathbb{F}(X)$ of the rank-2 symmetricspace $X={\rm SO}_0(2,2)/{\rm SO}(2)\times{\rm SO}(2)$. We introducecross-ratios on the torus in order to parametrise the ${\rmPSL}(2,\mathbb{R})^2$ configuration space of quadruples of pairwise distinctpoints in $\mathbb{T}$ and define a natural M\"obius structure in $\mathbb{T}$and therefore to $\mathbb{F}(X)$ and $\partial_\infty AdS^3$ as well.

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