The L^p –diameter of the group of area-preserving diffeomorphisms of s²

We show that for each p > 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, quasimorphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map X₄(CP¹) →_M0;4 š CP¹ n {∞; 0; 1} from the configuration space of 4 points on CP¹ to the moduli space of complex rational curves with 4 marked points.