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## Are these Riemann Surfaces biholomorphic?

Differential Geometry
Tsakanikas Nickos
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### Are these Riemann Surfaces biholomorphic?

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.