Geometry and topology of Fuchsian character varieties

Question

Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\mathbb H^2$ by under the action of a co-compact Fuchsian group. If $p=q=4g$ for $g \geq 2$, then the Fuchsian group can be taken to be the surface group, $\Sigma_g$. I'm interested in tessellations generated by co-comact Fuchsian groups, $\Gamma$. The problem from condensed matter physics I'm thinking about mathematically necessitates the study of character varieties $$C_{\Gamma,n}=\text{Hom}_{\text{Irr}}(\Gamma,\operatorname{SU}(n))$$

Most information I have come across is for the special case where $\Gamma = \Sigma_g$. For instance, the discussion after equation (1.1) in this paper states $C_{\Gamma,n}$ is a smooth manifold. Naturally, I have the following questions:

• What is known about the geometry of $C_{\Gamma,n}$ when $\Gamma$ is not a surface group? Is it a smooth manifold? Complex algebraic variety? Scheme?
• This paper linked above works with integrating functions on $C_{\Gamma,n}$ using the Atiyah-Bott-Goldman measure. Is there an analog of it for a co-compact Fuchsian group?
• Can anything be said under more assumptions on $\Gamma$?

The problem I am working on necessitates the study of $C_{\Gamma,n}$ when $\Gamma$ is a co-compact Fuchsian group. If there are no general answers to the above questions, can we make any "perturbative" argument in the case near $\Gamma$ is a "near-surface group." For instance, only one of the generators might be a elliptic element.

Answer 1

Let $\Gamma< PSL(2,\mathbb R)$ (not $SL(2,\mathbb R$)!) be a cocompact Fuchsian group. Given a Lie group $G$ one defines

1. The representation variety $R(\Gamma, G)=Hom(\Gamma,G)$. One further defines $R_0(\Gamma,G)\subset R(\Gamma,G)$ as the (open) subset consisting of homomorphisms $\rho$ such that the coadjoint representation $Ad^*\circ \rho$ of $\Gamma$ on the dual space of the Lie algebra ${\mathfrak g}$ of $G$ has no fixed vectors. (In the case $G=SU(n)$ this is a weaker assumption than irreducibility that you want to impose.)

If $G$ is compact (more generally, reductive), there is a natural invariant nondegenerate bilinear form on ${\mathfrak g}$ making coadjoint representation isomorphic to the adjoint representation. If $G$ is compact, one defines further quotients $$ X_0(\Gamma, G)\subset X(\Gamma,G) $$ obtained by taking quotients of $R(\Gamma, G), R_0(\Gamma, G)$ by the action of inner automorphism group of $G$. Informally, these are known as character varieties. (An algebraic geometer would object, but you are a physicist, so you would not care about this terminological abuse.)

Theorem. If $G$ is a Lie group, then $R_0(\Gamma, G)$ is a smooth manifold. (Ok, another terminological abuse here.) If $G$ is a compact Lie group then $X_0(\Gamma, G)$ is a smooth manifold as well.

You can find a proof in

Weil, André, Remarks on the cohomology of groups, Ann. Math. (2) 80, 149-157 (1964). ZBL0192.12802.

2. Regarding the symplectic structure. Let $\pi$ denote the fundamental group of a compact surface with boundary $S$, with $n$ oriented boundary circles $c_1,...,c_n$; I will identify these loops with corresponding elements $c_1,...,c_n\in \pi$ (well-defined up to conjugation). Let $G$ be a Lie group and $C_1,...,C_n$ are conjugacy classes in $G$. Given this, one defines the relative representation variety $$ R(\pi, G; C_1,...,C_n)=\{\rho\in Hom(\pi, G) | \rho(c_i)\in C_i, i=1,...,n\}. $$ Similarly, one defines $R_0(\pi, G; C_1,...,C_n)$ and, assuming $G$ is compact, one also defines relative character varieties: $$ X_0(\pi, G; C_1,...,C_n)\subset X(\Gamma,G; C_1,...,C_n). $$ Weil's theorem applies to these as well. It turns out that relative character varieties also have natural symplectic structures, see for instance

Jeffrey, Lisa C., Symplectic forms on moduli spaces of flat connections on 2-manifolds, Kazez, William H. (ed.), Geometric topology. 1993 Georgia international topology conference, August 2–13, 1993, Athens, GA, USA. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 2(pt.1), 268-281 (1997). ZBL0904.57009.

One similarly defines relative character varieties if each $C_i$ is a finite union of conjugacy classes, these are just disjoint unions of relative character varieties defined above. How does this relate to Fuchsian groups? Each cocompact Fuchsian group $\Gamma$ has a presentation where one starts with a group $\pi$ as above and adds relators $c_i^{p_i}=1$, $i=1,...,n$. Now, given a Lie group $G$, define finite unions of conjugacy classes in $G$, $C_1,...,C_n$, where $C_i$ consists of all elements of (finite) order dividing $p_i$. Then, assuming that $G$ is compact, we obtain natural isomorphisms (in any sense of the word) $$ R(\pi, G; C_1,...,C_n)\cong R(\Gamma, G),..., X_0(\pi, G; C_1,...,C_n)\cong X_0(\Gamma, G). $$ Thus, Jeffrey's symplectic form yields a symplectic form (of finite volume) on $X_0(\Gamma, G)$.

Many other things are known about geometry and topology of $X(\pi, G; C_1,...,C_n)$ and, hence, $X(\Gamma, G)$, but you probably do not have enough background (yet) for this discussion.