  For fixed g and T we show that finiteness of the set of affine equivalence
classes of flat surfaces of genus g whose Veech groups contain a cusp of
hyperbolic co-area less than T. We obtain new restrictions on Veech groups: we
show that any non-elementary Veech group can appear only finitely many times in
a fixed stratum, that any non-elementary Veech group is of finite index in its
normalizer, and that the quotient of the upper half plane by a non-lattice
Veech group contains arbitrarily large embedded disks. These are proved using
the finiteness of the set of affine equivalence classes of flat surfaces of
genus g whose Veech group contains a hyperbolic element with eigenvalue less
than T.
