Arithmetic Veech sublattices of SL\((2,\mathbf {Z})\). (English) Zbl 1244.32009

Ellenberg, Jordan S.; McReynolds, D. B.

Duke Math. J. 161, No. 3, 415-429 (2012).

A Teichmüller curve in the moduli space \(\mathcal{M}_{g,[n]}\) of genus \(g\) Riemann surfaces with \(n\) punctures \(S_{g,n}\) is a holomorphic curve \(f: C\to \mathcal{M}_{g,[n]}\) which is generically one-to-one and a local isometry with respect to the Kobayashi metric. In 2009, C. T. McMullen [Math. Res. Lett. 16, No. 4, 647–649 (2009; Zbl 1187.32010)] has proved that every Teichmüller curve has a model as an algebraic curve over \(\overline{\mathbb{Q}}\); in this paper the authors prove that, conversely, every algebraic curve over \(\overline{\mathbb{Q}}\) is birationally equivalent over \(\mathbb{C}\) to a Teichmüller curve (that can even be chosen in \(\mathcal{M}_{g'}\) for \(g'\) large enough, where \(\mathcal{M}_{g'}\) is the moduli space of genus \(g'\) Riemann surfaces with no punctures).
This result is a consequence of the main (group-theoretical) theorem of this paper. A Veech group is the stabilizer in the mapping class group \(\mathrm{Mod}(S_{1,1})\cong SL(2,\mathbb{Z})\) of the conjugacy class of a finite-index subgroup in the fundamental group of \(S_{1,1}\), the torus with one puncture. Then the authors prove that every finite-index subgroup of the level 2 principal congruence subgroup \(\Gamma(2)\subset SL(2,\mathbb{Z})\) containing \(\{\pm1\}\) is a Veech group.

Reviewer: Marco Abate (Pisa)

MSC:

32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Keywords:

Teichmüller curve; Veech groups

Citations:

Zbl 1187.32010

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