The Root Conjecture predicts that every pseudo-Anosov diffeomorphism of a closed surface has Teichmüller approximate th roots for all . In this paper, we replace the Teichmüller topology by the heights-widths topology – that is induced by convergence of tangent quadratic differentials with respect to both the heights and widths functionals – and show that every pseudo-Anosov diffeomorphism of a closed surface has heights-widths approximate th roots for all .
La Conjecture de la Racine prévoit que chaque difféomorphisme pseudo-Anosov d’une surface fermée a une racine ième approximative de Teichmüller pour tout . Dans cet article, on remplace la topologie de Teichmüller par la topologie hauteur-longueur – celle qui est induite par la convergence des différentielles quadratiques tangentes relativement aux fonctionnelles hauteurs et longueurs simultanément – et on prouve que chaque difféomorphisme pseudo-Anosov d’une surface fermée a une racine ième approximative hauteur-longueur pour tout .
Keywords: Teichmuller space, pseudo-Anosov diffeomorphism, root conjecture
Mot clés : espace de Teichmüller, difféomorphisme pseudo-Anosov, conjecture de la racine
@article{AIF_2009__59_4_1413_0, author = {Gendron, T. M.}, title = {Approximate roots of {pseudo-Anosov} diffeomorphisms}, journal = {Annales de l'Institut Fourier}, pages = {1413--1442}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {4}, year = {2009}, doi = {10.5802/aif.2469}, zbl = {1179.30044}, mrnumber = {2566966}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2469/} }
TY - JOUR AU - Gendron, T. M. TI - Approximate roots of pseudo-Anosov diffeomorphisms JO - Annales de l'Institut Fourier PY - 2009 SP - 1413 EP - 1442 VL - 59 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2469/ DO - 10.5802/aif.2469 LA - en ID - AIF_2009__59_4_1413_0 ER -
%0 Journal Article %A Gendron, T. M. %T Approximate roots of pseudo-Anosov diffeomorphisms %J Annales de l'Institut Fourier %D 2009 %P 1413-1442 %V 59 %N 4 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2469/ %R 10.5802/aif.2469 %G en %F AIF_2009__59_4_1413_0
Gendron, T. M. Approximate roots of pseudo-Anosov diffeomorphisms. Annales de l'Institut Fourier, Volume 59 (2009) no. 4, pp. 1413-1442. doi : 10.5802/aif.2469. http://archive.numdam.org/articles/10.5802/aif.2469/
[1] Determinant bundles, Quillen metrics and Mumford isomorphisms over the universal commensurability Teichmüller space, Acta Math., Volume 176 (1996) no. 2, pp. 145-169 | DOI | MR | Zbl
[2] Roots in the mapping class group (arXiv:math/0607278) | Zbl
[3] Automorphisms of Surfaces After Nielsen and Thurston, London Mathematical Society Student Texts 9, Cambridge University Press, Cambridge, 1988 | Zbl
[4] Cohomology with bounds, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 389-395 | MR | Zbl
[5] Travaux de Thurston sur les surfaces, Séminaire Orsay. Astérisque 66–67, 1991 (Société Mathématique de France, Paris) | MR
[6] Roots, symmetries and conjugacy of pseudo Anosov mapping classes (arXiv:0710.2043)
[7] The Theory of Matrices, 1 & 2, AMS Chelsea Publishing, Providence, RI, 1998 | MR | Zbl
[8] Teichmüller Theory and Quadratic Differentials, John Wiley and Sons, 1987 (Inc., New York) | MR | Zbl
[9] The Ehrenpreis conjecture and the moduli-rigidity gap, Complex manifolds and hyperbolic geometry (Guanajuato, 2001) 311, Contemp. Math., Amer. Math. Soc., Providence, RI (2002), pp. 207-229 | MR | Zbl
[10] Measured lamination spaces for surfaces, from the topological viewpoint, Topology Appl., Volume 30 (1988) no. 1, pp. 63-88 | DOI | MR | Zbl
[11] Quadratic differentials and foliations, Acta Math., Volume 142 (1979) no. 3, 4, pp. 221-274 | DOI | MR | Zbl
[12] Random ideal triangulations and the Weil-Petersson distance between finite degree covers of punctured Riemann surfaces (arXiv:0806.2304v1)
[13] The asymptotic geometry of Teichmüller space, Topology, Volume 19 (1980) no. 1, pp. 23-41 | DOI | MR | Zbl
[14] Dense geodesics in moduli space, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Princeton Univ. Press, Princeton, N.J., Ann. of Math. Stud. 97 (1981), pp. 417-438 | MR | Zbl
[15] Amenability, Poincaré series and quasiconformal maps, Invent. Math., Volume 97 (1989) no. 1, pp. 95-127 | DOI | EuDML | MR | Zbl
[16] A construction of pseudo Anosov homeomorphisms, Trans. Amer. Math. Soc., Volume 310 (1988) no. 1, pp. 179-197 | DOI | MR | Zbl
[17] Bounds on least dilatations, Proc. Amer. Math. Soc., Volume 113 (1991) no. 2, pp. 443-450 | DOI | MR | Zbl
[18] Combinatorics of Train Tracks., Annals of Mathematics Studies, 25, Princeton University Press, Princeton, NJ, 1992 | MR | Zbl
[19] Quadratic Differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 5, Springer-Verlag, Berlin, 1984 | MR | Zbl
[20] The Geometry and Topology of Three-Manifolds., Princeton University Notes (unpublished), 1979
[21] On the geometry and dynamics of diffeomorphisms of surfaces., Bull. Amer. Math. Soc., Volume (N.S.) 19 (1988) no. 2, pp. 417-431 | DOI | MR | Zbl
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