The L^p-diameter of the group of area-preserving diffeomorphisms of S²

2016/04/30 by Michael Brandenbursky, Egor Shelukhin · 1 citation
Mathematics · Physics and Astronomy · #math.GT #math-ph #math.DG #math.DS #math.MP #math.SG

paper · pdf · doi:10.2140/gt.2017.21.3785

published as Geom. Topol. 21 (2017) 3785-3810 · 19 pages, 1 figure; supersedes arXiv:1304.7037; small changes in exposition

arxiv created 2016/07/07 · arxiv updated 2018/03/16

Abstract

We show that for each p ≥ 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 X₄(ℂ P¹) → M_0,4 ≅ ℂ P¹ ∖ \∞,0,1\ from the configuration space of 4 points on ℂ P¹ to the moduli space of complex rational curves with 4 marked points.

Citations

• On the large-scale geometry of the L^p-metric on the symplectomorphism group of the two-sphere

Cited by

• The Schwarz-Milnor lemma for braids and area-preserving diffeomorphisms