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I will start learning about character varieties. I need to learn about Teichmuller spaces and how to consider them as components of "some character variety".

Can someone recommend some textbooks or papers, please? Thanks.

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William Goldman has proven that the components of the character variety of the fundamental group of a surface $S$ into $PSL_2(\mathbb{R})$ are in one-to-one correspondence with the induced Euler classes of the representations (which by the Milnor-Wood inequality are $\leq |\chi(S)|$), with the maximal and minimal Euler classes corresponding to copies of Teichmuller space. Have a look at his paper and references therein.

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