Mapping class group dynamics on Aff()-characters.
Annales de l'Institut Fourier, Volume 66 (2016) no. 2, p. 729-751
We prove that in genus bigger than 2, the mapping class group action on Aff()-characters is ergodic. This implies that almost every representation π 1 SAff() is the holonomy of a branched affine structure on S, where S is a closed orientable surface of genus g2.
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.
Received : 2014-09-15
Revised : 2015-02-05
Accepted : 2015-09-10
Published online : 2016-02-17
DOI : https://doi.org/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
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {2},
     year = {2016},
     pages = {729-751},
     doi = {10.5802/aif.3024},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2016__66_2_729_0}
}
Mapping class group dynamics on $\mathrm{Aff}(\mathbb{C})$-characters.. Annales de l'Institut Fourier, Volume 66 (2016) no. 2, pp. 729-751. doi : 10.5802/aif.3024. https://aif.centre-mersenne.org/item/AIF_2016__66_2_729_0/

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

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

[3] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton University Press, Princeton, NJ, Princeton Mathematical Series, Tome 49 (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), Tome 151 (2000) no. 2, pp. 625-704 | Article

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

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

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

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

[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, Birkhäuser Boston Inc., Boston, MA, Modern Birkhäuser Classics (2009), xxviii+467 pages (Reprint of the 2001 edition) | Article

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

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

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

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