5 edited tags
4 fixed typo.

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 $\kappa=exp(d_Teich)$ 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 $Teichmuller map ? A somewhat detailed explanation would be appreciated, thanks. 3 added 118 characters in body; edited body; added 15 characters in body Hello, I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved$ \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) $. I know we can define it via teichmuler Teichmuler metric, but they seems to have not defined it in this chapter ? So, how much structure about the set$ 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.