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$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$I studied the interpretation of Teichmuller space as a representation space for surface groups in $\PSL(2,\mathbb{R})$.

Now I am planning to study the representations of surface group into a Lie group. So, I am planning to read the book Lectures on representations of surface groups by F. Labourie. As far I understood that the above said book deals with differential geometric notions like connections, curvature, etc.

Also I think that above said F. Labourie's book "Lectures on representations of surface groups" is all about to investigate the local and global geometry of representation space, also involves questions about complex and symplectic structures. Also in the last chapter the author discusses a remarkable formula that gives the symplectic volume of the space of representations when $G$ is compact.

My first question is all about some notes or books where I can study throughly and rigorously about the symplectic structure on a manifold. Please provide references.

$\textbf{UPDATE:}$ I have found that the book "Introduction to Smooth Manifolds" by John M. Lee contains a chapter (last) on symplectic manifolds. I will learn about the symplectic structure on a manifold from that book

Now my second question is about the prospects or future directions of representations of surface groups. Maybe I am asking for interaction between

  1. Lectures on representations of surface groups and Hyperbolic manifolds and Kleinian groups.
  2. Lectures on representations of surface groups and Anti-de Sitter geometry.
  3. I have heard that Lectures on representations of surface froups has a lot of interactions with geometric structures (thanks to this answer). Does Introduction to Hitchin representations play a strong key role in the relation between Lectures on representations of surface groups and geometric structures?
  4. Do the geometric properties of representations of surface groups in $\SL(2,\mathbb{R})$ and $\SL(2,\mathbb{C})$ (or in general $\SL(n, \mathbb{R})$ strongly involve study of hyperbolic manifolds?
  5. Are there any relations or interactions between Lectures on representations of surface groups and geometric structures on low dimensional manifolds, in particular hyperbolic and anti-de Sitter $3$-manifolds?

Actually I am looking for further directions to study the surface group representations. Please advise. Thanks in advance.

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  • $\begingroup$ I recommend the book of Hofer for symplectic geometry and symplectic invariants, Hamiltonian dynamics. amazon.com/… $\endgroup$ Jan 18, 2023 at 12:10

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