What is Higher Teichmuller Theory?

Question

I am interested in representation of surface group. So, I am studying the book Lectures on Representations of Surface Groups by Francois Labourie. The main goal of representation of surface groups is to study the representation spaces of surface groups into (semi-simple) Lie groups.

We define representation variety of the fundamental group $\pi_{1}(S)$ of a closed connected surface $S$ of genus $g$ greater than $2$, with values in a Lie group $G$ as follows: $$\text{Rep} \big(\pi_{1} (S), G\big) := \hom \big(\pi_{1}(S), G\big) / G$$ where $G$ acts on $\hom (\pi_{1}(S), G)$ by conjugation.

Now, Teichmuller space is a connected component of the representation variety $\hom(\pi_1(S), PSL(2, \mathbb{R}))/PSL(2, \mathbb{R})$ - this is where higher Teichmuller theory takes it starting point. Instead of focussing on group homomorphisms of $\pi_1(S)$ into $PSL(2\mathbb{R})$, we replace $PSL(2, \mathbb{R})$ by a simple Lie group $G$ of higher rank such as $PSL(n, \mathbb{R})$, $n \geq 3$ or $Sp(2n, \mathbb{R})$, $n \geq 2$, and consider the representation variety $\hom(\pi_1(S), G)/G$. Therefore we make the following definition

$\textbf{Definition:}$ A higher Teichmuller space is a subset of $\hom(\pi_1(S), G)/G$, which is a union of connected components that consist entirely of discrete and faithful representations.

I have heard some terms of higher Teichmuller Theory such as Hitchin components, spaces of maximal representations, and Anosov representations. My question is how to study higher Teichmuller Theory and the above said terms? Please advise me about a roadmap of higher Teichmuller Theory.

And also how the Anosov representations is connected to the Margulis spacetimes (a noncompact complete Lorentz flat $3$-manifold $E /\Gamma$ with a free holonomy group $\Gamma$ of rank $g$, $g \geq 2$)? So, is there any bridge between higher Teichmuller Theory and Lorentzian geometry (in particular, $\textbf{anti de Sitter geometry}$)?

Please advise.

Answer 1

First of all, you should be working with the character variety $$\text{Rep} \big(\pi_{1} (S), G\big) := \hom \big(\pi_{1}(S), G\big) // G$$ which is the algebro-geometric quotient (the quotient that you wrote typically is not a variety). But this is a minor issue.

Here are references to few survey papers on the HTT (Higher Teichmuller Theory) and Anosov representations (according to all common definitions of HTT, representations in its context there should be Anosov, which is something that your definition is lacking).

[1] Burger, Marc; Iozzi, Alessandra; Wienhard, Anna, Higher Teichmüller spaces: from SL(2,R) to other Lie groups. Handbook of Teichmüller theory. Vol. IV, 539–618, IRMA Lect. Math. Theor. Phys., 19, Eur. Math. Soc., Zürich, 2014.

[2] Guichard, Olivier, Groupes convexes-cocompacts en rang supérieur [d'après Labourie, Kapovich, Leeb, Porti,…]. (French) [Higher-rank convex cocompact groups [based on Labourie, Kapovich, Leeb, Porti,…]] Astérisque No. 414, Séminaire Bourbaki. Vol. 2017/2018. Exposés 1136–1150 (2019), Exp. No. 1138, 95–123.

[3] Guichard, Olivier; Wienhard, Anna, Positivity and higher Teichmüller theory. European Congress of Mathematics, 289–310, Eur. Math. Soc., Zürich, 2018.

[4] Kapovich, Michael; Leeb, Bernhard Discrete isometry groups of symmetric spaces. Handbook of group actions. Vol. IV, 191–290, Adv. Lect. Math. (ALM), 41, Int. Press, Somerville, MA, 2018.

[5] Wienhard, Anna, An invitation to higher Teichmüller theory. Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, 1013–1039, World Sci. Publ., Hackensack, NJ, 2018.

Few more things: Hitchin components, spaces of maximal representations and positive representations of surface groups provide examples of HTT-components. I will not give any definitions since these are nontrivial and you will find these in the above references.

Margulis space times (MST) are a variation on the same theme where the group in question is free and not a surface group, spaces of corresponding representations do not form a connected component and the target group is not semisimple. My suggestion is to focus on the references I gave and ignore MST, this already will keep you busy for a long time.

See also my answers here, here, and answers to this MO question.

Lastly, you absolutely need a real advisor to help you to navigate in all this...