no. 2
Mapping class group dynamics on Aff(ℂ)-characters.
[Dynamique du groupe modulaire sur les caractères affines]
Ghazouani, Selim1
1 DMA - ÉNS 45 rue d’Ulm 75230 Paris Cedex 05 (France)
Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 729-751.

Nous prouvons dans cet article qu’en genre plus grand que deux, l’action du groupe modulaire sur les caractères affines est ergodique. Un corollaire de ce résultat est que presque toute représentation du groupe fondamental de S dans le groupe affine complexe est l’holonomie d’une structure affine branchée sur S, où S est une surface fermée orientable de genre plus grand que deux.

We prove that in genus bigger than 2, the mapping class group action on Aff(ℂ)-characters is ergodic. This implies that almost every representation π 1 S⟶Aff(ℂ) is the holonomy of a branched affine structure on S, where S is a closed orientable surface of genus g≥2.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3024
Classification : 22D40, 20F39, 57M05
Keywords: ergodic theory, mapping class group, Torelli group, character variety, complex affine group, complex branched affine structure
Mot clés : théorie ergodique, groupe modulaire, groupe de Torelli, variété de caractères, groupe affine complex, structure affine branchée

Ghazouani, Selim 1

1 DMA - ÉNS 45 rue d’Ulm 75230 Paris Cedex 05 (France) 
@article{AIF_2016__66_2_729_0,
     author = {Ghazouani, Selim},
     title = {Mapping class group dynamics on $\mathrm{Aff}(\mathbb{C})$-characters.},
     journal = {Annales de l'Institut Fourier},
     pages = {729--751},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {2},
     year = {2016},
     doi = {10.5802/aif.3024},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3024/}
}
TY  - JOUR
AU  - Ghazouani, Selim
TI  - Mapping class group dynamics on $\mathrm{Aff}(\mathbb{C})$-characters.
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 729
EP  - 751
VL  - 66
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3024/
DO  - 10.5802/aif.3024
LA  - en
ID  - AIF_2016__66_2_729_0
ER  - 
%0 Journal Article
%A Ghazouani, Selim
%T Mapping class group dynamics on $\mathrm{Aff}(\mathbb{C})$-characters.
%J Annales de l'Institut Fourier
%D 2016
%P 729-751
%V 66
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3024/
%R 10.5802/aif.3024
%G en
%F AIF_2016__66_2_729_0
Ghazouani, Selim. Mapping class group dynamics on $\mathrm{Aff}(\mathbb{C})$-characters.. Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 729-751. doi : 10.5802/aif.3024. https://aif.centre-mersenne.org/articles/10.5802/aif.3024/

[1] Chueshev, V. V. Cohomological Gunning fibration and the Torelli group, Sibirsk. Mat. Zh., Volume 31 (1990) no. 3, p. 198-203, 219 | DOI

[2] Deligne, P.; Mostow, G. D. Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math., Volume 63 (1986), pp. 5-89 | DOI

[3] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012, xiv+472 pages

[4] Gallo, Daniel; Kapovich, Michael; Marden, Albert The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. (2), Volume 151 (2000) no. 2, pp. 625-704 | DOI

[5] Goldman, William M. The symplectic nature of fundamental groups of surfaces, Adv. in Math., Volume 54 (1984) no. 2, pp. 200-225 | DOI

[6] Goldman, William M. Ergodic theory on moduli spaces, Ann. of Math. (2), Volume 146 (1997) no. 3, pp. 475-507 | DOI

[7] Haupt, Otto Ein Satz über die Abelschen Integrale 1. Gattung, Math. Z., Volume 6 (1920) no. 3-4, pp. 219-237 | DOI

[8] Jørgensen, Troels On discrete groups of Möbius transformations, Amer. J. Math., Volume 98 (1976) no. 3, pp. 739-749 | DOI

[9] Kapovich, Michael Periods of abelian differentials and dynamics (https://www.math.ucdavis.edu/~kapovich/EPR/fla.pdf)

[10] Kapovich, Michael Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2009, xxviii+467 pages (Reprint of the 2001 edition) | DOI

[11] Moore, Calvin C. Ergodicity of flows on homogeneous spaces, Amer. J. Math., Volume 88 (1966), pp. 154-178 | DOI

[12] Pickrell, Doug; Xia, Eugene Z. Ergodicity of mapping class group actions on representation varieties. I. Closed surfaces, Comment. Math. Helv., Volume 77 (2002) no. 2, pp. 339-362 | DOI

[13] Veech, William A. Flat surfaces, Amer. J. Math., Volume 115 (1993) no. 3, pp. 589-689 | DOI

[14] Zorich, Anton Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437-583 | DOI

Cité par Sources :