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 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 $g,h$ 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)