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Is there a residually finite finitely presented infinite groups $G$, such that there are only finitely many irreducible linear representation $\rho: G \to \text{GL}(n,\mathbb{C})$ with $\rho(G)$ finite (updated) up to conjugacy, for any fixed integer $n>0$?

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    $\begingroup$ Yes, using superrigidity, many lattices satisfy this, for instance $\mathrm{SL}_2(\mathbf{Z}[\sqrt{2}])$. $\endgroup$
    – YCor
    Nov 18, 2020 at 16:01
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    $\begingroup$ @YCor There are lots groups $G$ (e.g., free abelian groups) with positive dimensional character variety $Hom(G, GL(n, \mathbb{C}))// GL(n, \mathbb{C})$, so there could be uncountably many representation up to conjugacy. I don't quite understand your last comment. $\endgroup$
    – Feng Hao
    Nov 18, 2020 at 16:32
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    $\begingroup$ OK, just to clarify: I gave you an example, but restricting to finite image representation, then you just get more examples. So, $\mathrm{SL}_2(\mathbf{Z}[\sqrt{2}])$ is an answer to the updated question. $\endgroup$
    – YCor
    Nov 18, 2020 at 16:37
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    $\begingroup$ I suppose that YCor, in his second comment, meant "countably many linear representation of finite image up to conjugacy"? $\endgroup$
    – Chris
    Nov 18, 2020 at 16:57
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    $\begingroup$ If I'm correct, a finitely generated group has the property of having only countably many finite-dimensional reps with finite image, up to conjugation, iff it has no infinite virtually abelian quotient (i.e., every finite index subgroup has finite abelianization). And if so, it has finitely many for each given dimension. $\endgroup$
    – YCor
    Nov 18, 2020 at 18:16

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