Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{SL}(2,\mathbb R)$. We can generate tessellations, especially [$\{p,q\} \;\text{tesellations}$][1] 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][2] 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.


  [1]: https://dmitrybrant.com/2007/01/24/hyperbolic-tessellations
  [2]: https://arxiv.org/pdf/2101.00252