Permutation of Periodic Points of Veech Surfaces in H(2)

We study how Weierstrass points of Veech surfaces in H (2), the stratum of Abelian differentials on Riemann surfaces in genus two with a single zero of order two, are permuted. These surfaces were classified by McMullen relying on two invariants: discriminant and spin. More precisely, given a Veech surface in 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 Dih<inf>4</inf>, if D ≡<inf>4</inf> 0; to Dih<inf>5</inf>, if D ≡<inf>8</inf> 5; and to Dih<inf>6</inf>, if D ≡<inf>8</inf> 1. More-over, these same groups arise when considering only Dehn multitwists of the affine group. © 2024, American Institute of Mathematical Sciences. All rights reserved.