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

Brandenbursky, Michael¹ ; Shelukhin, Egor²

¹ Department of Mathematics, Ben-Gurion University, Beer-Sheva, Israel
² Department of Mathematics and Statistics, University of Montréal, Montréal, Canada

Geometry & topology, Tome 21 (2017) no. 6, pp. 3785-3810.

Résumé

We show that for each $p\ge 1$, 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 $X_4\left(ℂP^1\right)\to {\mathcal{ℳ}}_{0,4}\cong ℂP^1\setminus \left\{\infty ,0,1\right\}$ from the configuration space of $4$ points on $ℂP^1$ to the moduli space of complex rational curves with $4$ marked points.

Détail

DOI : 10.2140/gt.2017.21.3785

Classification : 20F65, 37E30, 53D99, 20F36, 57M07, 57R50, 57S05
Keywords: L^p-metrics, area-preserving diffeomorphisms, braid groups, quasimorphisms, cross-ratio, configuration space, quasi-isometric embedding

Bibliographie

[1] V Arnol’D, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier Grenoble 16 (1966) 319

[2] V I Arnol’D, The cohomology ring of the colored braid group, Mat. Zametki 5 (1969) 227

[3] V I Arnol’D, B A Khesin, Topological methods in hydrodynamics, 125, Springer (1998) | DOI

[4] A Banyaga, The structure of classical diffeomorphism groups, 400, Kluwer (1997) | DOI

[5] M Benaim, J M Gambaudo, Metric properties of the group of area preserving diffeomorphisms, Trans. Amer. Math. Soc. 353 (2001) 4661 | DOI

[6] M Bestvina, K Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002) 69 | DOI

[7] J S Birman, On braid groups, Comm. Pure Appl. Math. 22 (1969) 41 | DOI

[8] M Brandenbursky, On quasi-morphisms from knot and braid invariants, J. Knot Theory Ramifications 20 (2011) 1397 | DOI

[9] M Brandenbursky, Quasi-morphisms and Lp–metrics on groups of volume-preserving diffeomorphisms, J. Topol. Anal. 4 (2012) 255 | DOI

[10] M Brandenbursky, J Kędra, On the autonomous metric on the group of area-preserving diffeomorphisms of the 2–disc, Algebr. Geom. Topol. 13 (2013) 795 | DOI

[11] M Brandenbursky, J Kędra, Quasi-isometric embeddings into diffeomorphism groups, Groups Geom. Dyn. 7 (2013) 523 | DOI

[12] M Brandenbursky, E Shelukhin, On the large-scale geometry of the Lp–metric on the symplectomorphism group of the two-sphere, preprint (2015)

[13] M Brandenbursky, E Shelukhin, On the Lp–geometry of autonomous Hamiltonian diffeomorphisms of surfaces, Math. Res. Lett. 22 (2015) 1275 | DOI

[14] E Calabi, On the group of automorphisms of a symplectic manifold, from: "Problems in analysis" (editor R C Gunning), Princeton Univ. Press (1970) 1

[15] D Calegari, scl, 20, Math. Soc. Japan (2009) | DOI

[16] J Crisp, B Wiest, Quasi-isometrically embedded subgroups of braid and diffeomorphism groups, Trans. Amer. Math. Soc. 359 (2007) 5485 | DOI

[17] D G Ebin, Geodesics on the symplectomorphism group, Geom. Funct. Anal. 22 (2012) 202 | DOI

[18] D G Ebin, J Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970) 102 | DOI

[19] D G Ebin, G Misiołek, S C Preston, Singularities of the exponential map on the volume-preserving diffeomorphism group, Geom. Funct. Anal. 16 (2006) 850 | DOI

[20] Y Eliashberg, T Ratiu, The diameter of the symplectomorphism group is infinite, Invent. Math. 103 (1991) 327 | DOI

[21] A Fathi, Transformations et homéomorphismes préservant la mesure : systémes dynamiques minimaux, PhD thesis, Université Paris-Sud, Orsay (1980)

[22] J M Gambaudo, É Ghys, Enlacements asymptotiques, Topology 36 (1997) 1355 | DOI

[23] J M Gambaudo, É Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems 24 (2004) 1591 | DOI

[24] J M Gambaudo, M Lagrange, Topological lower bounds on the distance between area preserving diffeomorphisms, Bol. Soc. Brasil. Mat. 31 (2000) 9 | DOI

[25] P De La Harpe, Topics in geometric group theory, Univ. of Chicago Press (2000)

[26] T Ishida, Quasi-morphisms on the group of area-preserving diffeomorphisms of the 2–disk via braid groups, Proc. Amer. Math. Soc. Ser. B 1 (2014) 43 | DOI

[27] M Kapovich, RAAGs in Ham, Geom. Funct. Anal. 22 (2012) 733 | DOI

[28] C Kassel, V Turaev, Braid groups, 247, Springer (2008) | DOI

[29] B Khesin, R Wendt, The geometry of infinite-dimensional groups, 51, Springer (2009) | DOI

[30] S H Kim, T Koberda, Anti-trees and right-angled Artin subgroups of braid groups, Geom. Topol. 19 (2015) 3289 | DOI

[31] A N Kolmogorov, S V Fomın, Introductory real analysis, Dover (1975)

[32] S G Krantz, H R Parks, Geometric integration theory, Birkhäuser (2008) | DOI

[33] D Mcduff, D Salamon, Introduction to symplectic topology, Oxford Univ. Press (1998)

[34] L Polterovich, Hofer’s diameter and Lagrangian intersections, Int. Math. Res. Not. 1998 (1998) 217 | DOI

[35] L Polterovich, The geometry of the group of symplectic diffeomorphisms, Birkhäuser (2001) | DOI

[36] L Polterovich, Floer homology, dynamics and groups, from: "Morse theoretic methods in nonlinear analysis and in symplectic topology" (editors P Biran, O Cornea, F Lalonde), NATO Sci. Ser. II Math. Phys. Chem. 217, Springer (2006) 417 | DOI

[37] P Py, Quasi-morphismes de Calabi et graphe de Reeb sur le tore, C. R. Math. Acad. Sci. Paris 343 (2006) 323 | DOI

[38] P Py, Quasi-morphismes et invariant de Calabi, Ann. Sci. École Norm. Sup. 39 (2006) 177 | DOI

[39] E Shelukhin, “Enlacements asymptotiques” revisited, Ann. Math. Qué. 39 (2015) 205 | DOI

[40] A I Shnirel’Man, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. 128(170) (1985) 82, 144

[41] A I Shnirel’Man, Generalized fluid flows, their approximation and applications, Geom. Funct. Anal. 4 (1994) 586 | DOI

[42] A Shnirelman, Microglobal analysis of the Euler equations, J. Math. Fluid Mech. 7 (2005) | DOI

[43] S Smale, Diffeomorphisms of the 2–sphere, Proc. Amer. Math. Soc. 10 (1959) 621 | DOI