The Lp –diameter of the group of area-preserving diffeomorphisms of s2

Michael Brandenbursky, Egor Shelukhin

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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(CP1) →M0;4 š CP1 n {∞; 0; 1} from the configuration space of 4 points on CP1 to the moduli space of complex rational curves with 4 marked points.

Original languageEnglish
Pages (from-to)3785-3810
Number of pages26
JournalGeometry and Topology
Issue number6
StatePublished - 31 Aug 2017


  • Area-preserving diffeomorphisms
  • Braid groups
  • Configuration space
  • Cross-ratio
  • L^p-metrics
  • Quasi-isometric embedding
  • Quasimorphisms

ASJC Scopus subject areas

  • Geometry and Topology


Dive into the research topics of 'The Lp –diameter of the group of area-preserving diffeomorphisms of s2'. Together they form a unique fingerprint.

Cite this