  We relate a number of results in the theory of flat surfaces to BPS spectra
of a class of 4d $\mathcal{N}=2$ supersymmetric quantum field theories arising
from M5 branes wrapped on Riemann surfaces -- $A_1$ class S theories. In
particular, we apply classic results of Eskin and Masur, which determine the
asymptotic growth of geodesic counts at large length on flat surfaces, as well
as more recent progress in the mathematics literature, to determine the large
mass asymptotics of the BPS spectra of a wide class of such theories at generic
points in the Coulomb branch.
