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, 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.

19 pages, 1 figure; supersedes arXiv:1304.7037; small changes in exposition

References in corpus (1)

• Anti-trees and right-angled Artin subgroups of braid groups

Cited by in corpus (2)

• Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics
• A short proof that the $L^p$-diameter of $Dif{f}_0(S,area)$ is infinite