Let $G$ be a group. Suppose for any general linear representation $\rho:G\to\mathrm{GL}(n)$,
$\rho$ must be trivial.


Question: Are there any characterizations or equivalent conditions for $G$?


Thanks for guidance.



Characterizations of groups whose general linear representations are all trivial
With no limit on $n$, Martin Bridson and I proved that this property is undecidable for finitely presented groups. See the reformulation of the main theorem on page 2 of [Bridson and Wilton - The triviality problem for profinite completions](https://arxiv.org/abs/1401.2273):



> 
> There is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite-dimensional linear representation (over any field).
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> 


This reformulation goes as Ian Agol mentions in the [comments](https://mathoverflow.net/questions/376943/characterizations-of-groups-whose-general-linear-representations-are-all-trivial#comment956001_376943): every representation is trivial if and only if the group has no non-trivial finite quotients.


On the other hand, for fixed $n$ such problems are decidable. See, for instance, [Algorithms determining finite simple images of finitely presented groups](https://arxiv.org/abs/1710.07183) by Bridson, Evans, Liebeck, and Segal.


