Finiteness results for Veech groups

Veech groups are stabilizers of flat surfaces which arise in
the study of billiards on rational polygons and Teichmuller theory. These
are discrete subgroups of SL(2,R) which could be lattices but might not be
(they are sometimes infinitely generated!), and are the topic of lots of
recent research. I will describe work with John Smillie in which we
characterize the Veech groups which are lattices from 3 points of view:
the dynamics of the straight flow on the flat surface (Veech dichotomy and
its variants); the dynamics of the associated flows on the moduli space of
flat surfaces; the geometry of the flat surface. We also provide new
restrictions in case the groups are not lattices.