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Twisted Teichmüller Curves pp 1–10Cite as

Introduction

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2104))

Abstract

This is the introduction and summarizes the results proven in the following chapters.

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Notes

  1. 1.

    This theorem is one of the main reasons why it makes sense to define the well-known curves F N and T N , which are in many cases unions of certain twisted diagonals (see e.g. [HV74] or [McM07]). Moreover H.-G. Franke and W. Hausmann gave explicit formulas for the volume of the curves F N and T N ([Hau80], Satz 3.10 and Korollar 3.11). See Sect. 2.5.1.

  2. 2.

    The Kobayashi metric d is the largest pseudo-metric on X D such that for all holomorphic maps \(f: \mathbb{D} \rightarrow X_{D}\) we have: d(f(x), f(y)) ≤ ρ(x, y) where ρ is the Poincaré metric on the unit disk \(\mathbb{D}\).

  3. 3.

    This is not completely correct: A Teichmüller curve might also be the projection of a \(\mathrm{SL}_{2}(\mathbb{R})\)-orbit of a half-translation surface (X, q) where q is a quadratic differential. We may however restrict to the case (X, ω) by the so-called double covering construction.

  4. 4.

    A Teichmüller curve in \(\mathcal{M}_{g}\) is called primitive if it does not arise by a covering construction from a Teichmüller curve in lower genus.

  5. 5.

    If D = 5 then there is also a second primitive Teichmüller curve given by the regular decagon (see [McM06b]).

  6. 6.

    As the Hilbert modular surface X D parametrizes all principally polarized Abelian surfaces with real multiplication by \(\mathcal{O}_{D}\) (Theorem 2.30) one gets indeed an embedding of the Teichmüller curve into X D .

  7. 7.

    More generally it is, of course, true that all twists of Kobayashi curves are again Kobayashi curves, but we are only concerned about twisted Teichmüller curves in these notes.

  8. 8.

    There is a recent preprint [Muk12] of R. Mukamel where he claims to give an algorithm to find the Veech group of any Veech surface.

  9. 9.

    To be more precise: While C X, D lies on X D for all D, this is not always the case for C S, D . The curves C S, D always do allow real multiplication by \(\mathcal{O}_{D}\) but are not principally polarized. Therefore, for some D the curve lies C S, D on a different Hilbert modular surface (see Chap. 7).

  10. 10.

    Goethe Universität Frankfurt am Main, Institut für Algebra und Geometrie, Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main

    E-mail address: weiss@math.uni-frankfurt.de

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Goethe Universität Frankfurt am Main, Institut für Algebra und Geometrie, Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main

E-mail address: weiss@math.uni-frankfurt.de

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Weiß, C. (2014). Introduction. In: Twisted Teichmüller Curves. Lecture Notes in Mathematics, vol 2104. Springer, Cham. https://doi.org/10.1007/978-3-319-04075-2_1

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