Are these Riemann Surfaces biholomorphic?

Differential Geometry
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Tsakanikas Nickos
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Are these Riemann Surfaces biholomorphic?

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Post by Tsakanikas Nickos »

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.
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