# Some basic questions about the proof of Teichmuller's uniqueness theorem

Hello ,

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.

1. 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 ?

2. 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 $X$ ?

3. ( 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 ?

1. How exactly can we think of a teichmuller map as a map from $QD(X)$, space of holomorphic quadratic differentials on $X$, to $Teich(S_g)$ ?

1. Finally, what is/are really good reference for this topic ?
• Teichmuller space is the space of marked Riemann surfaces, and the homeomorphism $\phi$ is the marking for the point $[(X,\phi)]$. – Autumn Kent Jan 17 '11 at 22:25
• Also, which version of the book are you using? Those page numbers don't match with the version I have. – Autumn Kent Jan 17 '11 at 22:25
• I searched the latest version for "projective class" and they don't seem to talk about projective classes of quadratic differentials at all, so maybe you should look at the latest version: math.utah.edu/~margalit/primer – Autumn Kent Jan 17 '11 at 22:31
• @ Richard Kent : I am using version 4.03, if you are interested, I can send you the PDF file. – Analysis Now Jan 17 '11 at 22:35
• Thanks, I didn't know there is a more modern version 5.0, may be I would use it, thanks ! – Analysis Now Jan 17 '11 at 22:36