## Background

Let $S_{g,n}$ be an oriented surface of genus $g$, with $n$ punctures. We explicitly prohibit the *non-hyperbolic* cases:

- $g=0$, $n=0,1,2$.
- $g=1$, $n=0$.

Let $\Gamma$ be the fundamental group of $S_{g,n}$; a group arising this way is called a **Fuchsian group** (as opposed to some authors, we don't require that $\Gamma$ comes with an embedding into $PSL_2(\mathbb{R})$).

Let $G$ be a complete reductive algebraic group over a field $k$. The **representation algebra** $Rep(\Gamma,G)$ is defined such that $k$-algebra maps
$$ Rep(\Gamma,G)\rightarrow R$$
are naturally equivalent to group maps
$$ \Gamma\rightarrow G(R)$$
The **representation variety** is then $X_{\Gamma,G}:=Spec(Rep(\Gamma,G))$, despite the fact that this scheme can be non-reduced and hence not really a 'variety'.

## Question

For arbitrary groups $\Gamma$, the scheme $X_{\Gamma,G}$ can be quite bad. It is non-reduced $G=PSL_2(\mathbb{C})$ and for $\Gamma$ some Artinian groups (Kapovich-Millson, 1999) or for $\Gamma$ the fundamental group of some 3-manifolds (Kapovich, 2001).

For $\Gamma=\mathbb{Z}^2=\pi_{1,0}$ (one of the excluded cases), the representation variety $X_{\mathbb{Z}^2,G}$ is the *commuting scheme* of $G$. The reducedness of the commuting scheme is still an open question.

Despite this, it seems like it might be possible there are general theorems about nice properties of $X_{\Gamma,G}$, when $\Gamma$ is Fuchsian. For example, if $g=0$, then $\pi_{0,n}=F_{n-1}$, the free group on $n-1$ generators. Then $X_{F_{n-1},G}=G^{n-1}$, which is a smooth variety.

For $\Gamma$ Fuchsian, is it known whether $X_{\Gamma,G}$ is

- reduced?
- normal?
- smooth?

I have a vested interest in their normality, but that seems like the question least likely to be addressed directly.