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Mirzakhani, Maryam; Wright, Alex

Full-rank affine invariant submanifolds. (English) Zbl 1435.32016

Duke Math. J. 167, No. 1, 1-40 (2018).
“The relation between translation surfaces and billiards is given via a procedure called unfolding, which associates a translation surface to each polygon whose angles are rational multiples of \(\pi\). In fact, many specific applications to billiards have been elusive, since the set of unfoldings of polygons has measure zero. Consequently, although almost every translation surface has a dense \(\mathrm{GL}(2,\mathbb{R})\) orbit, the \(\mathrm{GL}(2,\mathbb{R})\) orbit closures of unfoldings have been unkown.”
In the paper under review, the authors prove that every \(\mathrm{GL}(2,\mathbb{R})\) orbit closure of a translation surface is a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. As an application, they show that the angles of any given polygon are multiples of \(\pi/3\), almost every polygon with the same angles unfolds to a translation surface whose orbit is dense in a connected component of the stratum. In addition, they also prove that there exist infinitely many rational triangles whose unfoldings have a dense \(\mathrm{GL}(2,\mathbb{R})\) orbit in a connected component of the stratum.
Reviewer: Jinhua Fan (Nanjing)

Cited in 2 Reviews
Cited in 16 Documents

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces

Keywords:

translation surfaces; billiards
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Full Text: DOI arXiv Euclid

References:

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