Hello,

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.