Permutation of periodic points of Veech surfaces in $\mathcal{H}(2)$

Abstract

We study how are permuted Weierstrass points of Veech surfaces in $\mathcal{H}(2)$, the stratum of Abelian differentials on Riemann surfaces in genus two with a single zero of order two. These surfaces were classified by McMullen relying on two invariants: discriminant and spin. More precisely, given a Veech surface in $\mathcal{H}(2)$ of discriminant $D$, we show that the permutation group induced by the affine group on the set of Weierstrass points is isomorphic to ${\mathrm{Dih}}_4$, if $D{\equiv }_{4}0$; to ${\mathrm{Dih}}_5$, if $D{\equiv }_{8}5$; and to ${\mathrm{Dih}}_6$, if $D{\equiv }_{8}1$. Moreover, these same groups arise when considering only Dehn multitwists of the affine group.

26 pages, 6 figures. Comments welcome!