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There are several puzzling questions on Riemann surface for me: Q.1 Definition of Riemann surface can be given in at least two ways: Def.1) it is a complex one dimensional manifold; Def.2) for each $a\in \mathbb{C}$, consider collection of germs at $a$, of analytic functions, and give a topology on it. Are these really equivalent definitions? or Def.2 is more general than Def.1?

Q.2 When we say a group $G$ is an automorphism group of a compact Riemann surface, how is the action? (for ex. what is description of action of of PSL(2,7) on a genus 3 Riemann surface? In the book of Thomas Breuer, I couldn't see any description of action of a group on a Riemann surface; he has given computational methods th to investigate groups.)

Q.3 The automorphisms of a compact Riemann surface can always be lifted to universal cover?

Q.4 If a group $G$ acts on a compact Riemann surface $X_g$, of genus $g$, then $X_g/G$ is also a compact Riemann surface of some genus $h$ and they $g,h$ are related by Riemann-Hurwitz formula. Can anyone suggest some good reference for this relation? (here, I would like to see this Riemann Hurwitz relation topologically; many books describe it using algebraic geometry techniques).

(I went through many books on Riemann surface for these questions; but not understood many things)

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# some questions on Riemann surface

There are several puzzling questions on Riemann surface for me: Q.1 Definition of Riemann surface can be given in at least two ways: Def.1) it is a complex one dimensional manifold; Def.2) for each $a\in \mathbb{C}$, consider collection of germs at $a$, of analytic functions, and give a topology on it. Are these really equivalent definitions? or Def.2 is more general than Def.1?

Q.2 When we say a group $G$ is an automorphism group of a compact Riemann surface, how is the action? (for ex. what is description of action of of PSL(2,7) on a genus 3 Riemann surface? In the book of Thomas Breuer, I couldn't see any description of action of a group on a Riemann surface; he has given computational methods th investigate groups.)

Q.3 The automorphisms of a compact Riemann surface can always be lifted to universal cover?

Q.4 If a group $G$ acts on a compact Riemann surface $X_g$, of genus $g$, then $X_g/G$ is also a Riemann surface of some genus $h$ and they are related by Riemann-Hurwitz formula. Can anyone suggest some good reference for this relation? (here, I would like to see this Riemann Hurwitz relation topologically; many books describe it using algebraic geometry techniques).

(I went through many books on Riemann surface for these questions; but not understood many things)