MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ?
share|cite|improve this question
Teichmuller space is the space of marked Riemann surfaces, and the homeomorphism $\phi$ is the marking for the point $[(X,\phi)]$. – Richard 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. – Richard 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: – Richard 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

Answer to your last question:

Quasiconformal Maps and Teichmüller Theory (Oxford Graduate Texts in Mathematics) [Hardcover] Alastair Fletcher (Author), Vladimir Markovic (Author)

share|cite|improve this answer
Thanks Dr Rivin, I hope that book has a pretty clear explanation . – Analysis Now Jan 17 '11 at 20:50

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.