The Lp–diameter of the group of area-preserving diffeomorphisms of S2

Brandenbursky, Michael; Shelukhin, Egor

doi:10.2140/gt.2017.21.3785

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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

DOI: 10.2140/gt.2017.21.3785

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, quasimorphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map $X_{4} (ℂ P^{1}) \to ℳ_{0, 4} ≅ ℂ P^{1} ∖ {\infty, 0, 1}$ from the configuration space of $4$ points on $ℂ P^{1}$ 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

References

Publication

Received: 8 July 2016

Accepted: 28 January 2017

Published: 31 August 2017

Proposed: Yasha Eliashberg

Seconded: Danny Calegari, Leonid Polterovich

Authors

Michael Brandenbursky
Department of Mathematics Ben-Gurion University Beer-Sheva Israel

Egor Shelukhin
Department of Mathematics and Statistics University of Montréal Montréal Canada

Volume 21, issue 6 (2017)