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A genus 2 characterisation of Veech surfaces


Speaker: Yitwah Cheung, San Francisco State University
Location: 2112 MSB
Start time: Wed, Feb 8 2006, 4:10PM

An abelian differential (X,omega) on a compact Riemann surface may be thought of geometrically as defining a flat metric with isolated conical singularities. Charts can be chosen away from the zeros so that the change of coordinates are of the form z -> z+c, so that not only is the Euclidean metric preserved, but parallel lines are sent to parallel lines. Hence, abelian differentials are sometimes referred to as translation surfaces. There is a nonholomorphic action of SL(2,R) on the moduli space of abelian differentials (given by post composing the natural charts phi:U->C=R2 by R-linear maps). For most translation surfaces (abelian differentials) the stabilizer is trivial. However, in some cases, the stabiliser is "large" in the sense that it forms a lattice in SL(2,R) i.e. a discrete subgroup of finite covolume. These surfaces may be thought of as natural generalisations of flat tori. Veech showed that the following hold for the dynamics on these surfaces: the translation flows in a given direction is minimal if and only if it is uniquely ergodic. In this talk, I will discuss the converse for genus 2 surfaces. This work is joint with Howard Masur.