some questions on Riemann surface - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T04:13:06Z http://mathoverflow.net/feeds/question/55023 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55023/some-questions-on-riemann-surface some questions on Riemann surface Martin David 2011-02-10T12:23:25Z 2011-02-11T11:51:17Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/55014/some-questions-on-riemann-surface" rel="nofollow">some questions on Riemann surface</a> </p> </blockquote> <p>There are several puzzling questions on Riemann surface for me: <strong>Q.1</strong> Definition of Riemann surface can be given in at least two ways: Def.1) <em>it is a complex one dimensional manifold</em>; Def.2) <em>for each $a\in \mathbb{C}$, consider collection of germs at $a$, of analytic functions, and give a topology on it</em>. Are these really equivalent definitions? or Def.2 is more general than Def.1?</p> <hr> <p><strong>Q.2</strong> 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.)</p> <hr> <p><strong>Q.3</strong> The automorphisms of a compact Riemann surface can always be lifted to universal cover?</p> <hr> <p><strong>Q.4</strong> 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).</p> <hr> <p>(I went through many books on Riemann surface for these questions; but not understood many things)</p>