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).


30F60 Teichmüller theory for Riemann surfaces
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