A good reference to learn Teichmuller Theory?

Question

I'm looking for a reference to start learning Teichmuller Theory.

Answer 1

For an introduction, I recommend Part 2 of the book "Primer on mapping class groups", by B. Farb and D. Margalit. It gives a quick glance into the definition of Teichmueller space in terms of hyperbolic metrics, conformal structures and also in terms of representations of $\pi_1(S)$ into $PSL_2(\mathbb{R})$. It discusses briefly quadratic differentials and the Teichmueller theorems.

If you want to get deeper into the theory of quasi-conformal maps and understand the complex structure of Teichmueller space, then the ultimate reference is J. Hubbard's book "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Vol. 1". It gives a very thorough treatment of Teichmueller theory, from the proof of the uniformization theorem that relates the definition in terms of conformal structures and hyperbolic structures, to quadratic differentials, the theory of quasi-conformal maps (with all the nitty gritty of its analytic and geometric definitions), the Bers embedding, the tangent space of Teichmueller space and its holomorphic structure and the Teichmueller theorems (existence and uniqueness of quasi-conformal maps). It also discusses the geometric viewpoint in terms of Fenchel-Nielsen coordinates, and gives a detailed proof of Wolpert's formula of the Weil-Petersson form in terms of these coordinates.

For even a broader scope of topics (but less self-contained and fewer details), I also recommend the succinct but excellent book "An Introduction to Teichmüller Spaces", by Y. Imayoshi and M. Taniguchi.