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If $\Gamma$ is a finitely generated group. Consider the representation $\mathrm{Rep}(\Gamma, \mathrm{SL}_2(\mathbb{C} )):=\mathrm{Hom}(\Gamma, \mathrm{SL}_2(\mathbb{C} ))$.

How can we show $\mathrm{Rep}(\Gamma, \mathrm{SL}_2(\mathbb{C} ))$ is an affine algebraic set? I know we can get the polynomials from the relations between the generators, but I do not know how to write them down.

Also if $\Gamma$ is not finitely generated, Is the representation space an affine algebraic set? If not, what makes it work in the finitely generated case?

Could someone help me, please?

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    It doesn't make sense to ask whether something which is _a priori_ a set is an affine algebraic set, in the same way that it doesn't make sense to ask whether something which is _a priori_ a set is a group. (It does make sense to ask whether a subset of an affine algebraic set is an algebraic subset, but $\text{Hom}(\Gamma, \text{SL}_2(\mathbb{C}))$ doesn't come with a canonical embedding into an affine algebraic set.) The best you can ask for is that there exists a canonical affine algebraic structure, but this is extra structure, not a property.2012-09-20
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    Anyway, I think this is a good exercise to do for yourself. Try the case $\Gamma = \mathbb{Z}$ first, then perhaps $\Gamma = F_n$, then perhaps $\Gamma = \mathbb{Z}^n$. If you get stuck then there is something more basic that you don't understand and you should be asking questions about that instead.2012-09-20

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