Fundamental domains and generators for lattice Veech groups

Abstract

The moduli space QMg of non-zero genus g quadratic differentials has a natural action of G D GL 2 .R/= ̋ ̇ 1 0 0 1 ̨ . The Veech group PSL.X; q/ is the stabilizer of .X; q/ 2 QMg in G. We describe a new algorithm for finding elements of PSL.X; q/ which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, we give the first explicit examples of generators and fundamental domains for non-arithmetic Veech groups where the genus of H=PSL.X; q/ is greater than zero. Mathematics Subject Classification (2010). 32G15, 30F30.