Isolation theorems for quadratic differentials

I will describe joint work with John Smillie giving finiteness
results for sets of lattice surfaces. A lattice surface is a pair $(S,q)$,
where $S$ is a compact orientable surface and $q$ is a translation structure
(or complex structure with a holomorphic 1-form) whose stabilizer in
$SL(2,{\bf R})$ is a lattice. Veech showed that non-arithmetic lattices may arise
and that billiards connected with lattice surfaces have nice dynamical
propertries (“Veech dichotomy”). We give concrete conditions for lattice
surfaces, and characterize them dynamically in terms of both the
corresponding billiard and the $SL(2,{\bf R})$ action. We also show that there is
a finite number of lattice surfaces with a fixed lower bound on the area
of any triangle, and that fixing the genus of $S$, there is only a finite
number with a fixed upper bound on the covolume.