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

Abstract

We show that for each $p \geq 1,$ 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 $X_4(\mathbb{C} P^1) \to \mathcal{M}_{0,4} \cong \mathbb{C} P^1 \setminus \{\infty,0,1\}$ from the configuration space of $4$ points on $\mathbb{C} P^1$ to the moduli space of complex rational curves with $4$ marked points.Comment: 19 pages, 1 figure; supersedes arXiv:1304.7037; small changes in expositio