#### Volume 21, issue 6 (2017)

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

### Michael Brandenbursky and Egor Shelukhin

Geometry & Topology 21 (2017) 3785–3810
##### 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 ${X}_{4}\left(ℂ\phantom{\rule{0.3em}{0ex}}{P}^{1}\right)\to {\mathsc{ℳ}}_{0,4}\cong ℂ\phantom{\rule{0.3em}{0ex}}{P}^{1}\setminus \left\{\infty ,0,1\right\}$ from the configuration space of $4$ points on $ℂ\phantom{\rule{0.3em}{0ex}}{P}^{1}$ to the moduli space of complex rational curves with $4$ marked points.

##### Keywords
L^p-metrics, area-preserving diffeomorphisms, braid groups, quasimorphisms, cross-ratio, configuration space, quasi-isometric embedding
##### Mathematical Subject Classification 2010
Primary: 20F65, 37E30, 53D99
Secondary: 20F36, 57M07, 57R50, 57S05