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

TL;DR

This paper classifies how the affine group of Veech surfaces in the stratum (2) permutes their Weierstrass points, revealing a connection between the discriminant and specific dihedral groups.

Contribution

It explicitly determines the permutation groups induced by the affine group on Weierstrass points based on the discriminant, linking algebraic invariants to geometric symmetries.

Findings

Permutation groups are dihedral, depending on discriminant congruences.

The groups are isomorphic to Dih_4, Dih_5, or Dih_6.

Dehn multitwists generate the same permutation groups.

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 $D \mathbin{\equiv_{4}} 0$; to $\mathrm{Dih}_5$, if $D \mathbin{\equiv_{8}} 5$; and to $\mathrm{Dih}_6$, if $D \mathbin{\equiv_{8}} 1$. Moreover, these same groups arise when considering only Dehn multitwists of the affine group.