In Frederick Gardiner's book Teichmuller Theory and Quadratic Differentials, P.27-28, Chapter 1 ) that dimension of $dim_RQD(X) = 6g-6+3m+2n $ ( by using Riemann-Roch theorem ). Now for open annulus $A$, $ g=0, m=2, n=0 $, we get $ dim_RQD(X)=0 $ ! I am a bit puzzled why it is zero ! (Should I define the genus of an open annulus to be zero ?)

For q.diffs $q$ on the annulus $A$, should we look at $ q=\phi(z)dz^2 $ when $\phi$ is a function on the annulus embedded in complex plane or should we lift it to upper half plane and consider the $ \phi(z) $ with $ \phi(z) = \phi(\gamma(z)) ({\gamma'(z)})^2$ for all $\gamma \in Deck(H/A) $ ? I guess the second approach makes more sense because it respects the hyperbolic geometric structure on $A$ as well ?


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