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

Abstract

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, quasimorphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map $X_4\left(ℂP^1\right)\to {\mathcal{ℳ}}_{0,4}\cong ℂP^1\setminus \left\{\infty ,0,1\right\}$ from the configuration space of $4$ points on $ℂP^1$ to the moduli space of complex rational curves with $4$ marked points.