mathimatikoi.org

Examine whether the following Riemann Surfaces are biholomorphic :

• $ \mathbb{H} \, / < z \mapsto z + 2 > $ and $ \mathbb{H} \, / < z \mapsto z + 3 > $.
• $ \mathbb{H} \, / < z \mapsto 2z > $ and $ \mathbb{H} \, / < z \mapsto 3z > $.

where by $ < z \mapsto \frac{az+b}{cz+d} > $ we mean the subgroup of $ Aut( \mathbb{H} ) \cong PSL(2,\mathbb{R}) $ generated by the specified element.