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I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved $ \omega : QD_1(X) -> Teich ( S_g) $ is proper, which, with continuity and uniqueness/injectivity would give the result after applying Brower's invariance of domain. I have a question about this proof and its notation.

First, to prove the continuity of $\omega$ , we need to have a topology on $ Teich(S_g) $. I know we can define it via Teichmuler metric, but they seems to have not defined it in this chapter ? So, how much structure about the set $ Teich(S_g) $ should I assume and exactly what topology are they using ?

They are actually trying to prove here that $ \kappa=exp(d_{Teich})$ [ still undefined so far ] is continuous. What did they mean by the symbol $ DF( \pi_1(S_g), PSL(2,R) ? $ I know the individual meanings of these notations, but I am not familiar with this symbol. What is $ F $ though ?

I also cannot follow " the marked Fundamental domains for these representations can be made K-quasiconformally equivalent for any $ K > 1$ by taking $Y'$ sufficiently close to $Y $ : what is a marked Fundamental domain, is it just any fundamental domain for the marked surface ? And how can they be made $ K $ q.c equivalent for any $ K> 1$ ?

Also, "By teichmullers uniqueness theorem, the infimum $\kappa(Y) $ is realized by some $K_h$ : why os it true for ALL $Y$ ? isn't it true for only the image of $\omega$ by Teichmuller uniqueness , because we are constructing a new surface from a given Riemann surface and a q.d. such that the identity map becomes $K q.c $ Teichmuller map ?

A somewhat detailed explanation would be appreciated, thanks.

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Can anyone suggest a good self-contained proof of Teichmuller's existence theorem which does not use heavy machinery ? ( I know Alistair and Markovich's book also has the proof, but I guess they develop and use some machinery . –  Analysis Now Feb 10 '11 at 17:58
I don't there is any low-tech proof of Teichmuller's existence theorem. However, there is an alternate proof using harmonic maps which some people find easier. An exposition can be found in Jost's book on compact Riemann surfaces. –  Andy Putman Feb 10 '11 at 20:47
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2 Answers

In chapter "Teichmuller Space" they have defined the notation $DF(\pi_1 (S_g),PSL(2,R)).$ This means The set of all discrete faithful representations of $\pi_1(S,g)$ into $PSL(2,R)$. Quotient of this space by $PGL(2,R)$ is in bijective correspondence with $Teich(S_g)$ and the topolology of the space is defined to be the topology of this quotient. $d_{Teich}$ has been defined in the first paragraph of Tichmuller extremal problem.

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