\(\mathrm{GL}_2\mathbb{R}\) orbit closures in hyperelliptic components of strata (Q2413075)

Given a pair \((X, \omega)\) of a compact Riemann surface \(X\) and holomorphic \(1\)-form \(\omega\), integrating \(\omega\) yields an atlas of charts from \(X\) (away from the zeros of \(\omega\)) to \(\mathbb C\). The chart around a point \(p_0\) is given by \[ z(p) = \int_{p_0}^p \omega, \] and in these charts \(\omega = dz\), and the transition maps are translations, justifying the term \textit{translation surface} for the pair \((X, \omega)\). The \(\mathbb R\)-linear action of \(\mathrm{GL}^+(2, \mathbb R)\) on \(\mathbb R^2\) allows one to define an action on the space of translation surfaces via postcomposition with these charts. Understanding orbits and orbit closures of these actions has applications to counting problems for closed trajectories for billiards, ergodic theory of billiards and interval exchange maps, and other interesting problems. An important class of surfaces is those with \textit{closed orbits}. By a theorem of Smillie, these coincide with the class of \textit{lattice surfaces}, i.e. surfaces (when normalized to have unit area) that have a stabilizer \(\mathrm{SL}(X, \omega)\) under the \(\mathrm{SL}(2, \mathbb R)\)-action which is a lattice in \(\mathrm{SL}(2, \mathbb R)\). The projections of these orbits to the moduli space of curves are known as \textit{Teichmüller curves}. More recently, pioneering work of Eskin-Mirzakhani, Eskin-Mirzakhani-Mohammadi, and Filip, showed that \(\mathrm{GL}^+(2, \mathbb R)\) orbit closures are defined by local linear equations in \textit{period coordinates} on the space of translation surfaces (here, period coordinates refer to the coordinates obtained by integrating \(\omega\) on a basis for homology relative to its singularities, that is, they are coordinates given by viewing \(\omega\) as an element of relative cohomology). Understanding what these orbits can be beyond this general result is a very important question. In the paper under review, the author resolves many important questions, in particular a conjecture of Mirzakhani, in the setting of \textit{hyperelliptic} strata of translation surface. The main results of the paper are about \textit{higher-rank} orbit closures, and how they arise. Namely, it is shown that higher-rank affine invariant submanifolds in hyperelliptic components arise exclusively from branched covering constructions; that is, the affine invariant submanifold consists of translation surfaces which cover a translation surface in a lower genus hyperelliptic component. As a corollary, there are only finite many algebraically primitive Teichmüller curves in all hyperelliptic components for genus greater than two.