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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

We show that for each p 1, the Lp–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(P1) 0,4P1 {,0,1} from the configuration space of 4 points on P1 to the moduli space of complex rational curves with 4 marked points.

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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
Received: 8 July 2016
Accepted: 28 January 2017
Published: 31 August 2017
Proposed: Yasha Eliashberg
Seconded: Danny Calegari, Leonid Polterovich
Michael Brandenbursky
Department of Mathematics
Ben-Gurion University
Egor Shelukhin
Department of Mathematics and Statistics
University of Montréal