Euler characteristics of Teichmüller curves in genus two

Bainbridge, Matt

doi:10.2140/gt.2007.11.1887

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Euler characteristics of Teichmüller curves in genus two

Matt Bainbridge

Geometry & Topology 11 (2007) 1887–2073

DOI: 10.2140/gt.2007.11.1887

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, ${SL}_{2} (ℝ)$ –invariant measure on the moduli space of Abelian differentials in genus two (previously calculated in unpublished work of Kontsevich and Zorich).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

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Publication

Received: 21 November 2006

Accepted: 12 July 2007

Published: 15 October 2007

Proposed: Shigeyuki Morita

Seconded: Benson Farb, Walter Neumann

Authors

Matt Bainbridge
Department of Mathematics University of Chicago 5734 S University Avenue Chicago, IL 60637 USA

Volume 11, issue 4 (2007)