A [classical result](http://www-users.math.umd.edu/%7Ewmg//fricke.pdf) of Fricke--Klein--Vogt from the late 1800s implies that the character variety $\chi\_\mathbb{C}$ associated to the free group $F\_2$ and the algebraic group $\mathrm{SL}\_2(\mathbb{C})$ is isomorphic to $\mathbb{C}^3$.


Question: What happens if we change $\mathbb{C}$ to a different (commutative unital) ring $R$? I.e., can one explicitly describe the character variety $\chi\_R$ associated to $F\_2$ and $\mathrm{SL}\_2$ over an arbitrary ring $R$? For which rings $R$, do we have $\chi\_R\simeq R^3$?


I suspect that $\chi\_\mathbb{R}$ is not isomorphic to $\mathbb{R}^3$ while $\chi\_{\mathbb{F}\_q}\simeq \mathbb{F}\_q^3$.



Character variety of the free group
Let $\pi$ be a free group of rank 2. The character variety of $SL\_{2,k}$-representations of $\pi$ is always isomorphic to affine 3 space $\mathbb{A}^3\_k$ *for any ring $k$*.


Let $A[\pi] = A[\pi]\_k$ denote the coordinate ring of $SL\_{2,k}\times SL\_{2,k}$, which we identify with $Hom(\pi, SL\_{2,k})$ by picking a basis of $\pi$. Thus
$$A[\pi] \cong k[\{a\_i,b\_i,c\_i,d\_i\}\_{i=1,2}]/(a\_id\_i-b\_ic\_i-1)\_{i=1,2}$$
Let $X\_i$ be the matrix variable $[[a\_i,b\_i],[c\_i,d\_i]]$. I claim that if we let $GL\_{2,k}$ act on $A[\pi]$ by simultaneous conjugation on matrices, then $A[\pi]^{GL\_{2,k}}$ is a polynomial ring in 3 variables.
$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\bZ}{\mathbb{Z}}$


This is treated in Brumfiel-Hilden's book "SL(2)-representations of finitely presented groups" (combine their Propositions 9.1(ii) and Proposition 3.5). Specifically 9.1(ii) says that for any ring $k$, the ring of invariants $A[\pi]^{GL\_{2,k}}$ is isomorphic to a certain ring $TH[\pi]$ (a subalgebra generated by "traces"), and Proposition 3.5 says that $TH[\pi]$ is in fact a polynomial ring over $k$ in the variables $A = \tr(X\_1)$, $B = \tr(X\_2)$, and $C = \tr(X\_1X\_2)$.


However, there is an issue with their statement - they actually claim that $k[A,B,C] = TH[\pi] = A[\pi]^{GL\_2(k)}$ (taking invariants by $GL\_2(k)$ instead of $GL\_{2,k}$). This cannot be true, since when $k$ is a finite field, $A[\pi]^{GL\_2(k)}$ is the invariant ring of a 6-dimensional algebra by a finite group, which must also be 6-dimensional, whereas $k[A,B,C]$ is visibly 3-dimensional. However, this is the only issue in their exposition. If you replace all instances of $GL\_2(k)$ by $GL\_{2,k}$, then their proof is fine.


I recently encountered this error when I tried to use their results in a paper of mine. Here are two ways to address their gap:


1. When $k = \bZ$ or $k$ is an infinite field, $GL\_2(k)$ is Zariski-dense in $GL\_{2,k}$, and hence in these cases their argument essentially works as is. You can bootstrap from these cases to the general case as follows. Since you know the case for $k = \mathbb{Z}$, you have $\bZ[A,B,C] = A[\pi]\_\bZ^{GL\_{2,\bZ}}$, so it suffices to check that taking invariants commutes with base change to any ring $k$. For any ring $k$, the universal coefficients theorem (Jantzen, I Proposition 4.18) gives an exact sequence
$$0\longrightarrow A[\pi]\_\bZ^{GL\_{2,\bZ}}\otimes\_\bZ k\longrightarrow A[\pi]\_k^{GL\_{2,k}}\longrightarrow Tor\_1^\bZ(H^1(GL\_{2,\bZ},A[\pi]\_\bZ),k)\longrightarrow 0$$
Thus it suffices to show that $H^1(GL\_{2,\bZ},A[\pi]\_\bZ)$ is $\bZ$-flat (in fact it is 0, but we don't need this). For this, it suffices to check vanishing of the Tor group when $k = \mathbb{F}\_p$ (for all $p$), and since $\overline{\mathbb{F}\_p}$ is faithfully flat over $\mathbb{F}\_p$, it suffices to check this when $k = \overline{\mathbb{F}\_p}$, but since we are assuming Brumfiel-Hilden's result over infinite fields, the exact sequence above gives us this Tor vanishing for $k = \overline{\mathbb{F}\_p}$, as desired.
2. Another approach is to put together pieces of their argument to make your own direct argument. Firstly, Brumfiel-Hilden's arguments for Proposition 3.5 are correct. Specifically, you can check that for any ring $k$, the map $k[x,y,z] \rightarrow A[\pi]^{GL\_{2,k}}$ gives an isomorphism onto the subalgebra $T[\pi]\subset A[\pi]$ generated by traces $\tr(X\_{i\_1}X\_{i\_2}\cdots X\_{i\_r})$ where each $i\_j\in\{1,2\}$ and $r\ge 1$ (injectivity is Prop 3.2, surjectivity is prop 1.7). The next piece you need is the correct statement of "The first fundamental theorem of invariant theory". Namely, you need to replace $GL\_2(k)$-invariants in their Proposition 9.2 (first part) with $GL\_{2,k}$-invariants. This is due to Donkin (*Invariants of several matrices*) when $k = \bZ$ or $k$ is an algebraically closed field, but again the same argument as above using the universal coefficient theorem extends this to all rings. This theorem says that if $A[2]$ denotes the polynomial ring on 8 variables representing two 2x2 matrices, then $A[2]^{GL\_{2,k}}$ ($GL\_2$ acting by simultaneous conjugation) is generated as a $k$-subalgebra by traces and determinants of arbitrary products of the universal matrices. Thus, to complete the argument it suffices to check that the surjective map $A[2]\rightarrow A[\pi]$ induces a surjection on $GL\_{2,k}$-invariants (note the determinants map to $1\in A[\pi]$). For this one can use a piece of Brumfiel-Hilden's argument (specifically in $\S9$, p97-98, "surjectivity of $\rho\_n$"), which is correct and holds over arbitrary rings $k$.


