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If you take an infinite simple group $G$ (say Thompson’s F) and put the discrete topology on it this will have your property when k is a finite field (since the image in $GL_n(k)$ will necessarily be trivial.

Say further that $G$ is finitely presented (as indeed for example Thompson's F is). Then you can generalize this to $\mathbb{C}$ as follows by following the techniques in this paper. Basically, a non-trivial homomorphism into $GL_n(\mathbb{C})$ can be turned into a homomorphism into $GL_n(\mathbb{F}_p)$ for some prime $p$. See theorem 3.4. There they use it to turn a $\mathbb{C}$ representation with non-commutative image into a $\mathbb{F}_p$ representation with non-commutative image, but you can do the same thing for "non-trivial" instead of "non-commutative" $\mathbb{C}$ representation. I will spell it out below.

If you take an infinite simple group $G$ (say Thompson’s $T$) and put the discrete topology on it this will have your property when k is a finite field (since the image in $GL_n(k)$ will necessarily be trivial.

Say further that $G$ is finitely presented (as indeed for example Thompson's $T$ is). Then you can generalize this to $\mathbb{C}$ as follows by following the techniques in this paper. Basically, a non-trivial homomorphism into $GL_n(\mathbb{C})$ can be turned into a homomorphism into $GL_n(\mathbb{F}_p)$ for some prime $p$. See theorem 3.4. There they use it to turn a $\mathbb{C}$ representation with non-commutative image into a $\mathbb{F}_p$ representation with non-commutative image, but you can do the same thing for "non-trivial" instead of "non-commutative" $\mathbb{C}$ representation. I will spell it out below.

Here is the idea. The key result it relies on is a very cool and useful theorem that says "solutions to systems of equations over $\mathbb{C}$ can be realized over large enough finite fields." Specifically, if $f_1(x_1, \dots, x_N), \dots f_m(x_1, \dots, x_N)$ is a system of polynomials with integer coefficients that has a solution over $\mathbb{C}$, then it also has a solution over $\mathbb{F}_p$ for some prime $p$. Edit: This was using a theorem cited in the attached paper. I replaced this with an easy to prove theorem that gives you a solution over a finite field (not necessarily prime order), proven below.

Next we want to construct a system of polynomials such that a solution to that system in a field $k$ will correspond to a $k$-representation.

At this point, a solution to the system over $k$ is exactly a $k$ representation that sends $x$ to a non-trivial element of $GL_n(k)$. We assumed such a representation exists for $\mathbb{C}$. Thus, the lemma proved below yields a representation for some finite field. But this is a contradiction, as we showed earlier that there are no representations over finite fields. Thus, there cannot be representations over $\mathbb{C}$ either.

The key result it relies on is a very cool and useful theorem that says "solutions to systems of equations over $\mathbb{C}$ imply over some finite field." Specifically, if $f_1(x_1, \dots, x_N), \dots f_m(x_1, \dots, x_N)$ is a system of polynomials with integer coefficients that has a solution over $\mathbb{C}$, then it also has a solution over $\mathbb{F}_p$ for some prime $p$. Edit: This was using a theorem cited in the attached paper. I replaced this with an easy to prove theorem that gives you a solution over a finite field (not necessarily prime order), proven below.

Next we want to construct a system of polynomials such that a solution to that system in a field $k$ will correspond to a $k$-representation.

At this point, a solution to the system over $k$ is exactly a $k$ representation that sends $x$ to a non-trivial element of $GL_n(k)$. We assumed such a representation exists for $\mathbb{C}$. Thus, the lemma proved below yields a representation for some finite field. But this is a contradiction, as we showed earlier that there are no representations over finite fields. Thus, there cannot be representations over $\mathbb{C}$ either.

With the theorems as stated in the above paper it seems to require we assume GRH, but I don't think that's required if we don't care about any upper bound on $p$.

Here is the idea. The key result it relies on is a very cool and useful theorem that says "solutions to systems of equations over $\mathbb{C}$ can be realized over large enough finite fields." Specifically, if $f_1(x_1, \dots, x_N), \dots f_m(x_1, \dots, x_N)$ is a system of polynomials with integer coefficients that has a solution over $\mathbb{C}$, then it also has a solution over $\mathbb{F}_p$ for some prime $p$.

At this point, a solution to the system over $k$ is exactly a $k$ representation that sends $x$ to a non-trivial element of $GL_n(k)$. We assumed such a representation exists for $\mathbb{C}$. Thus, the theorem cited above yields a representation for $\mathbb{F}_p$. But this is a contradiction, as we showed earlier that there are no $\mathbb{F}_p$ representations. Thus, there cannot be representations over $\mathbb{C}$ either.

Here is the idea. The key result it relies on is a very cool and useful theorem that says "solutions to systems of equations over $\mathbb{C}$ can be realized over large enough finite fields." Specifically, if $f_1(x_1, \dots, x_N), \dots f_m(x_1, \dots, x_N)$ is a system of polynomials with integer coefficients that has a solution over $\mathbb{C}$, then it also has a solution over $\mathbb{F}_p$ for some prime $p$. Edit: This was using a theorem cited in the attached paper. I replaced this with an easy to prove theorem that gives you a solution over a finite field (not necessarily prime order), proven below.

At this point, a solution to the system over $k$ is exactly a $k$ representation that sends $x$ to a non-trivial element of $GL_n(k)$. We assumed such a representation exists for $\mathbb{C}$. Thus, the lemma proved below yields a representation for some finite field. But this is a contradiction, as we showed earlier that there are no representations over finite fields. Thus, there cannot be representations over $\mathbb{C}$ either.

Edit: After some reflection I realized we can do without the results of the paper in this context by proving an easier version of a similar theorem.

Lemma. Fix any prime $p$. If $f_1(x_1, \dots, x_N), \dots f_m(x_1, \dots, x_N) \in \mathbb{Z}[x_1, \dots, x_N]$ have a common zero over $\mathbb{C}$, then they have a common zero over some finite field of characteristic $p$.

Proof.

Consider the ideal $I_p = (f_1, \dots, f_m) \subseteq \mathbb{F}_p[x_1, \dots, x_N]$.

1. First suppose this ideal is trivial (i.e., equal to $\mathbb{F}_p[x_1, \dots, x_N]$). Then there exist some $g_1, \dots, g_m \in \mathbb{F}_p[x_1, \dots, x_N]$ with $$ g_1 f_1 + \dots + g_m f_m = 1 \mod p $$ Now let $G_i \in \mathbb{Z}[x_1, \dots, x_N]$ be any polynomial which when reduced mod $p$ is $g_i$. Then $$ G_1 f_1 + \dots + G_m f_m = 1 + a p $$ for some $a \in \mathbb{Z}$. Then we have $$ \frac{G_1}{1 + ap} f_1 + \dots + \frac{G_m}{1+ ap} f_m = 1 $$ which means the ideal $(f_1, \dots, f_m) \subseteq \mathbb{C}[x_1, \dots, x_N]$ is trivial. But that means there can be no solution to the $f_i$ over $\mathbb{C}$, which is a contradiction to our assumption.

2. So, it must be the case that $I_p$ is not trivial. Then, by the weak nullstellensatz, the $f_i$ have a common solution $\alpha_1, \dots, \alpha_m$ in $\overline{\mathbb{F}_p}$, the algebraic closure of $\mathbb{F}_p$. Since each of the $\alpha_i$ is contained is some finite extension of $\mathbb{F}_p$, we can take a finite extension $k$ of $\mathbb{F}_p$ that contains all the $\alpha_i$. So we are done: $k$ is our finite field of characteristic $p$ which has a solution $(\alpha_1, \dots, \alpha_m)$ to our system of equations.