I was studying the proof of Teichmuller's uniqueness theorem from the note/book " A Primer on Mapping Class Groups " by Farb-Margalit and I got struck at a couple of points, mainly because I am new to the subject. It would be great if you answer some of my questions :
Here are some questions I might like to ask :
Let $X$ be a closed Riemann surface, $ S_g $ be topological closed oriented genus g surface, $q_X $ be a holomorphic quadratic differential on X.
Why [($X$, $\phi $)] should be an element in $Teich(S_g$) ? ( P. 291 of Farb-Margalit ) . I mean why should we put phi there ? Different phi just gives different laminations of X , so why should we put it there ?
What exactly is meant by " projective classes of $ q_X $" ? ( P 291 of Farb-Margalit ) Why does it give a tangent direction in the tangent space $T_X(Teich(S_g))$ of $Teich(S_g)$ at $X$ ?
( P. 292 of Farb-Margalit ) Given $X, q_X, K > 1$, we can cook up a new closed Riemann surface $Y$ such that there is a Teichmller map f with initial QD $q_X$, terminal QD $q_Y$, and stretch factor K > 1 in the following way :
First puncture $X$ at the zeros of $ q_X$, take natural coordinate chart , and then compose with the affine map f (x,y) = ($ \sqrt(K)x, \sqrt(1/K)y $). But then the new transition maps become $f o \(z_1) o(z_2inverse )o f inverse $[ sorry about bad notation ], where o means composition . This new map is NOT holomorphic, although the rest except the f-parts is holomorphic . So how do we get a Riemann surface structure ?
How exactly can we think of a teichmuller map as a map from $QD(X)$, space of holomorphic quadratic differentials on $X$, to $ Teich(S_g) $ ?
- Finally, what is/are really good reference for this topic ?