Surface Group Representations

Question

I am interested in Hyperbolic Geometry. I studied hyperbolic surfaces, the space of all marked hyperbolic structures on a surface (also known as the Teichmuller space of the surface), and the interpretation of Teichmuller space as a representation space for surface groups in $PSL(2,\mathbb{R})$.

Now I want to study the representation 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.

Now my question is as follows.

The hyperbolic geometry appear for representations when the target group is either $PSL(2, {\mathbb R})$ or $PSL(2, {\mathbb C})$.

Is the topic representation of surface group into Lie group (in particular, F.Labourie's book) connected with hyperbolic geometry topics (such as geometric structures, hyperbolic $3$ manifolds, Lorentzian geometry etc)? Please advise.

Answer 1

1. Hyperbolic geometry appears in the context of representations (of arbitrary groups!) with discrete image, when the target group is $(,1)$ (essentially, the isometry group of the hyperbolic $$-space). Thus, there is no reason to limit yourself to isometries of hyperbolic plane and hyperbolic 3-space, as in your post.

2. Regarding the question in your last paragraph:

Your list "geometric structures, hyperbolic 3-manifolds, Lorentzian geometry" is a hodge-podge of topics. (I do not know what "etc" is in your question, hence, I will simply ignore it.)

Let's take these items one at a time.

(i) Geometric structures: Yes, absolutely. For instance, one of the early examples of Hitchin representations of surfaces groups was $\pi_1(S)\to PGL(3, {\mathbb R})$. These were precisely the holonomy representations of convex real-projective structures on the surface $S$. Labourie's book lays out foundations for this topic but does not go into it.

(ii) 3-manifolds: One has to understand these broadly by allowing compact manifolds with boundary. See chapter 7 of the book.

(iii) Lorentzian geometry: Yes, but this is not covered in the book. The target Lie group $Isom({\mathbb R}^{n,1})$ in this case is the semidirect product of ${\mathbb R}^{n+1}$ and $O(n,1)$. This group is not semisimple, hence the connection is much more tenious. But, representations of (punctured) surface groups to $Isom({\mathbb R}^{2,1})$ correspond to infinitesimal deformations of hyperbolic structures on surfaces (see chapter 5 of the book).