Event Type:

Department Colloquium

Date/Time:

Tuesday, November 2, 2004 - 07:00

Location:

Kidd 364

Local Speaker:

Abstract:

A region bounded by a Euclidean polygon can be viewed as a billiard table: the billiard ball paths are the straight-line paths, with angle-preserving reflections off of the sides (consider vertices as ``pockets"). A standard unfolding process associates to each such table a surface; if the interior angles at the vertices are rational multiples of $\pi$, then a finite genus oriented closed topological surface results. But, more is true: due to the complex structure on the original plane, the topological surface has a natural complex structure as well --- it is a Riemann surface, $X$. In a fundamental result, W. Veech showed in 1989 that ergodic properties of the billiard table dynamics are reflected in certain aspects of $X$.

In fact, the unfolding process identifies a 1-form, $\omega$ on $X$. There is a natural action of $\text{SL}(2, \mathbb R)$ on the collection of all 1-forms on the Riemann surfaces of a fixed genus (I will briefly present this action). The Veech result is expressed in terms of the stabilizer of $\omega$. In 1992, Veech asked if this stabilizer must always be a finitely generated group. In 2004, Pascal Hubert and I showed that this is not the case. I will sketch our proof. Time allowing, I will indicate a recent response of C. McMullen to the question as well.