Square-tiled surfaces
Samuel Lelievre, Warwick Mathematics Institute

Abelian differentials on Riemann surfaces can be seen as flat surfaces
with cone-type singularities. Square-tiled surfaces, or torus covers,
are integer or rational points of their moduli spaces, and as such,
these combinatorial objects are a very good tool to investigate these
moduli spaces. As an illustration, the volumes of all moduli spaces
of abelian differentials were computed using countings of square-tiled
surfaces, which have quasimodular forms as generating functions.

A notable feature of moduli spaces of abelian differentials is their
natural SL(2,R) action, which seems to exhibit the same rigidity as
the action of unipotent flows on homogeneous spaces. There are a
variety of stabilisers for this action, from trivial to infinitely
generated. Those of square-tiled surfaces are finite-index subgroups
of SL(2,Z), and most of them seem to be noncongruence subgroups.

Another problem of interest is that of counting closed geodesics on
flat surfaces. Such countings have quadratic growth rates and almost
always precise quadratic asymptotics. In particular, square-tiled
surfaces have explicit quadratic asymptotics, which in some cases
allow to retrieve generic asymptotics.