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What is a good reference for Teichmuller spaces of punctured surfaces $S_{g,n}$ where $n>0$?

I am looking for a reference where there is the correct statement and or proof of say the Bers embedding, Royden's theorem and the proof that $T(S_{g,n})$ is homeomorphic to an open ball in $\mathbb{R}^{3g-3+n}$.

Have Teichmuller curves in $M_{g,n}$ been explored much?

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  • $\begingroup$ In fact, it is hard to find a textbook on Teichmuller theory that excludes punctured surfaces. I cannot think of any. $\endgroup$
    – Moishe Kohan
    Commented Apr 2 at 16:48
  • $\begingroup$ An Introduction to Teichmüller Spaces by Imayoshi and Taniguchi only deals with the closed case. $\endgroup$
    – Sam Nead
    Commented Apr 3 at 7:45

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Fred Gardiner's book Teichmüller theory and quadratic differentials is a good reference. He (a) deals with the punctured case (called finite analytical type: see the first page of Chapter 2), (b) covers the Bers embedding (see Section 5.4), (c) covers a version of Royden's theorem (see Section 9.2), and (d) proves that Teichmüller space is an open ball (see Section 6.7).

In answer to your other question: yes, Teichmüller disks in moduli space have been explored by many mathematicians. Much is known, and much is unknown! McMullen has recently written a survey, with references to other works: the bibliography is six and a half pages.

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  • $\begingroup$ Thanks! In his survey only Teichmuller curves in $M_g$ have been discussed not in $M_{g,n}$ for $n>1$. $\endgroup$
    – Chitrabhanu
    Commented Apr 3 at 5:26
  • $\begingroup$ The punctured case reduces to the closed case. One way to see this is via the locus of quadratic differentials associated to a Teichmüller curve. These are all either (a) global squares of abelian differentials or (b) not. In case (a) we can remove all punctures and get the associated Teichmüller curve in $M_g$ (the genus does not change). In case (b) we form the (possibly branched) double cover coming from the failure of the differentials to be a global square and then we remove the punctures. This gives the associated Teichmüller curve in $M_{g'}$ (the genus does change). $\endgroup$
    – Sam Nead
    Commented Apr 3 at 8:47
  • $\begingroup$ If my answer has answered your question, you should consider accepting it - you do this by checking the "tick mark" (which will then turn green). $\endgroup$
    – Sam Nead
    Commented Apr 3 at 8:48
  • $\begingroup$ So none of the Teichmuller curves in $M_{g,n}$ come from a quadratic differential with a simple pole at some of the punctures? How does one see this? $\endgroup$
    – Chitrabhanu
    Commented Apr 5 at 12:14
  • $\begingroup$ That is not what I said. What I said is this: every Teichmuller curve in $M_{g,n}$ is (canonically) associated to some Teichmuller curve in $M_{g'}$ where $g' \geq g$ depends on the original given curve. $\endgroup$
    – Sam Nead
    Commented Apr 5 at 17:15

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