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Rank 2 affine manifolds in genus \(3\). (English) Zbl 1455.30037

A translation surface is a pair of a Riemann surface \(X\) together with a holomorphic 1-form \(\omega\). Fixing the topological type of \(X\), there is a moduli space \(\mathcal{H}\) of translation surfaces, which admits an action of \(\mathrm{SL}_2(\mathbb{R})\). The moduli space is stratified according to the type of zeroes of the 1-form, and this stratification is \(\mathrm{SL}_2(\mathbb{R})\)-invariant.
In general, however, closures of orbits can be more complicated than full strata, and classifying orbit closures is an extremly complicated and rich question. C. T. McMullen [Ann. Math. (2) 165, No. 2, 397–456 (2007; Zbl 1131.14027)] obtained such a classification in genus 2. More recently, seminal work of A. Eskin and M. Mirzakhani [Publ. Math., Inst. Hautes Étud. Sci. 127, 95–324 (2018; Zbl 1478.37002)], A. Eskin et al. [Ann. Math. (2) 182, No. 2, 673–721 (2015; Zbl 1357.37040)], S. Filip [Ann. Math. (2) 183, No. 2, 681–713 (2016; Zbl 1342.14015)] shows that in general orbit closures are affine submanifolds (in suitable natural charts of the space of translation structures) with finite ergodic \(\mathrm{SL}_2(\mathbb{R})\)-invariant measure, and also have the structure of quasi-projective varieties.
A full classification of such orbit closures in higher genus is beyond the scope of current methods, but partial results exist. In particular, A. Wright [Geom. Topol. 19, No. 1, 413–438 (2015; Zbl 1318.32021)] defined a complexity measure for orbit closures, called the cylinder rank, which varies between 1 and the genus \(g\) of the surface \(X\). Surfaces whose orbit closures have cylinder rank 1 are geometrically very constrained (they are completely periodic). In certain strata of genus 3, rank 2 orbit closures have also been classified before [the second author and A. Wright, Geom. Funct. Anal. 24, No. 4, 1316–1335 (2014; Zbl 1303.30039); the first author et al., J. Eur. Math. Soc. (JEMS) 18, No. 8, 1855–1872 (2016; Zbl 1369.37044); Geom. Topol. 20, No. 5, 2837–2904 (2016; Zbl 1370.32006)].
The central result of this article completes this classification of rank-2 orbit closures in genus 3 (Theorem 1.1). Combining this result with work of [M. Mirzakhani and A. Wright, Duke Math. J. 167, No. 1, 1–40 (2018; Zbl 1435.32016)] yields a description of all \(\mathrm{GL}^+(2,\mathbb{R})\)-orbit closures in genus 3 (Theorem A).

MSC:

30F60 Teichmüller theory for Riemann surfaces
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References:

[1] David Aulicino and Duc-Manh Nguyen. Rank two affine submanifolds in H(2,2) andH(3,1).Geom. Topol., 20(5):2837-2904, 2016. MR3556350, Zbl 1370.32006. · Zbl 1370.32006
[2] David Aulicino, Duc-Manh Nguyen, and Alex Wright. Classification of higher rank orbit closures inHodd(4).J. Eur. Math. Soc. (JEMS), 18(8):1855-1872, 2016. MR3518480, Zbl 1369.37044. · Zbl 1369.37044
[3] Paul Apisa. GL2Rorbit closures in hyperelliptic components of strata. Duke Math. J., 167(4):679-742, 2018. MR3769676. · Zbl 1436.32053
[4] David Aulicino. Affine manifolds and zero Lyapunov exponents in genus 3. Geom. Funct. Anal., 25(5):1333-1370, 2015. MR3426056, Zbl 1357.37078. · Zbl 1357.37078
[5] Alex Eskin and Maryam Mirzakhani. Invariant and stationary measures for the SL(2,R) action on moduli space.Publ. Math. Inst. Hautes ´Etudes Sci., 127:95-324, 2018. MR 3814652, Zbl 06914160. · Zbl 1478.37002
[6] Alex Eskin, Maryam Mirzakhani, and Amir Mohammadi. Isolation, equidistribution, and orbit closures for the SL(2,R) action on moduli space.Ann. of Math. (2), 182(2):673-721, 2015. MR3418528, Zbl 1357.37040. · Zbl 1357.37040
[7] Simion Filip. Splitting mixed Hodge structures over affine invariant manifolds.Ann. of Math. (2), 183(2):681-713, 2016. MR3450485, Zbl 1342.14015. · Zbl 1342.14015
[8] Benson Farb and Dan Margalit.A primer on Mapping Class Groups, volume 49 ofPrinceton Mathematical Series. Princeton University Press, 2012. Zbl 1245.57002. · Zbl 1245.57002
[9] Giovanni Forni. A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle.J. Mod. Dyn., 5(2):355-395, 2011. With an appendix by Carlos Matheus. MR2820565 (2012f:37061), Zbl 1238.30031. · Zbl 1238.30031
[10] Maxim Kontsevich and Anton Zorich. Connected components of the moduli spaces of Abelian differentials with prescribed singularities.Invent. Math., 153(3):631-678, 2003. MR2000471 (2005b:32030), Zbl 1087.32010. · Zbl 1087.32010
[11] Erwan Lanneau. Connected components of the strata of the moduli spaces of quadratic differentials.Ann. Sci. ´Ec. Norm. Sup´er. (4), 41(1):1-56, 2008. MR2423309 (2009e:30094), Zbl 1161.30033. · Zbl 1161.30033
[12] Erwan Lanneau, Duc-Manh Nguyen, and Alex Wright. Finiteness of Teichm¨uller curves in non-arithmetic rank 1 orbit closures.Amer. J. Math., 139(6):1449-1463, 2017. MR3730926, Zbl 1382.32011. · Zbl 1382.32011
[13] Curtis T. McMullen. Prym varieties and Teichm¨uller curves.Duke Math. J., 133(3):569-590, 2006. Zbl 1099.14018. · Zbl 1099.14018
[14] Curtis T. McMullen. Dynamics of SL2(R) over moduli space in genus two. Ann. of Math. (2), 165(2):397-456, 2007. MR2299738 (2008k:32035), Zbl 1131.14027. · Zbl 1131.14027
[15] Curtis T. McMullen, Ronen E. Mukamel, and Alex Wright. Cubic curves and totally geodesic subvarieties of moduli space. Ann. of Math. (2) 185(3):957-990, 2017. MR3664815, Zbl 06731862. · Zbl 1460.14062
[16] Maryam Mirzakhani and Alex Wright. The boundary of an affine invariant submanifold.Invent. Math., 209(3):927-984, 2017. MR3681397, Zbl 1378.37069. · Zbl 1378.37069
[17] Maryam Mirzakhani and Alex Wright. Full-rank affine invariant submanifolds.Duke Math. J., 167(1):1-40, 2018. MR3743698. · Zbl 1435.32016
[18] Duc-Manh Nguyen and Alex Wright. Non-Veech surfaces inHhyp(4) are generic.Geom. Funct. Anal., 24(4):1316-1335, 2014. MR3248487, Zbl 1303.30039. · Zbl 1303.30039
[19] John Smillie and Barak Weiss. Minimal sets for flows on moduli space. Israel J. Math., 142:249-260, 2004. MR2085718 (2005g:37067), Zbl 1052.37025. · Zbl 1052.37025
[20] Alex Wright. The field of definition of affine invariant submanifolds of the moduli space of abelian differentials.Geom. Topol., 18(3):1323-1341, 2014. MR3254934, Zbl 1320.32019. · Zbl 1320.32019
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