Characterizations of groups whose general linear representations are all trivial

MathOverflow Asked by QSH on December 7, 2020

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

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

Thanks for guidance.

One Answer

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:

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).

This reformulation goes as Ian Agol mentions in the comments: 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 by Bridson, Evans, Liebeck, and Segal.

Correct answer by HJRW on December 7, 2020

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