The $L^p$-diameter of the group of area-preserving diffeomorphisms of $S^2$

We show that for each $p\ge 1,$ the $L^p$-metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of Shnirelman from 1985. Our methods extend to yield stronger results on the large-scale geometry of the corresponding metric space, completing an answer to a question of Kapovich from 2012. Our proof uses configuration spaces of points on the two-sphere, quasi-morphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map $X_4($\mathbb{C}$ P^1) \to $\mathcal{M}$_{0,4} \cong $\mathbb{C}$ P^1 \setminus {\infty,0,1}$fromtheconfigurationspaceof$4$pointson$$\mathbb{C}$ P^1$tothemodulispaceofcomplexrationalcurveswith$4$ marked points.