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

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, quasimorphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map X4(CP 1 ) → M0,4 ∼ = CP 1 \ {∞, 0, 1} from the configuration space of 4 points on CP 1 to the moduli space of complex rational curves with 4 marked points.

PFH spectral invariants on the two-sphere and the large scale geometry of Hofer's metric

“…14.2]; it is mentioned as one of the motivations behind the influential article of Polterovich and Shelukhin [PS16, Sec. 1.3]; and it is highlighted in several articles such as [Py08,EPP12,KS18,BS17].…”

A short proof that the $L^p$-diameter of $Diff_0(S,area)$ is infinite

“…The last case which was left was the sphere, and it was showed recently by Brandenbursky and Shelukhin [3] that in this case the diameter is as well infinite, and moreover Diff 0 (S, area) contains quasi-isometrically embedded right-angled Artin groups and R m for each natural m. Their arguments use ideas from [5], however, using intersection numbers in the case of the sphere is not that straightforward and requires considerably more work.…”

Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics

“…For M being either a (two-dimensional) surface with boundary or a closed surface of genus g ≥ 2 this conjecture has been shown to be true by Eliashberg and Ratiu [23] for any L p metric, p ≥ 1. The case of the torus and the (significantly more complicated) case of S 2 were proved in [14], thus proving infinite diameter for any closed two dimensional surface with respect to the L p metric. So far, to the best of our knowledge, the analogue of Shnirelman's question regarding boundedness (unboundedness resp.)…”