Volume 11, issue 4 (2007)

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Euler characteristics of Teichmüller curves in genus two

Matt Bainbridge

Geometry & Topology 11 (2007) 1887–2073
 arXiv: math/0611409
Abstract

We calculate the Euler characteristics of all of the Teichmüller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves are naturally embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmüller curve is proportional to the Euler characteristic of the Hilbert modular surface on which it lies.

The idea is to use techniques from algebraic geometry to calculate the fundamental classes of these Teichmüller curves in certain compactifications of the Hilbert modular surfaces. This is done by defining meromorphic sections of line bundles over Hilbert modular surfaces which vanish along these Teichmüller curves.

We apply these results to calculate the Siegel–Veech constants for counting closed billiards paths in certain L-shaped polygons. We also calculate the Lyapunov exponents of the Kontsevich–Zorich cocycle for any ergodic, ${SL}_{2}\left(ℝ\right)$–invariant measure on the moduli space of Abelian differentials in genus two (previously calculated in unpublished work of Kontsevich and Zorich).

Keywords
Teichmüller curves, Hilbert modular surfaces, billiards, Lyapunov exponents
Mathematical Subject Classification 2000
Primary: 32G15, 37D50
Secondary: 11F41, 30F30