Perhaps it will be easier to use complex numbers and unitary representations.

There is a joint publication with Minkowski.

Isn't it the chain rule?

The theory of representations of the symmetric groups is well developed.

A virtually residually finite group is always residually finite.

Normally hyperbolic means that no power has an iinvariant proper subspace, parabolic means that some power has an invariant proper subspace but the element is not torsion and elliptic means torsion. Since in dim $\ge 3$ every matrix has invariant $2-$dim subspace, you won't have hyperbolic matrices if $d\ge3$.

By definition, a problem for which the question about complexity makes sense must be (equivalent to) a decision problem.

@StefanKohl: For most of these groups the statement is obvious (I assume the "core" means normal core).

There are not that many perfect groups of order less than $100000$.

OK ${}{}{}{}{}{}$

Doesn't it prove that the example does not exist if dim>5? I have not read the paper but I remember that they proved the analog of Cannon's conjecture for big enough dimension.

How about this paper? Ferry, Steve; Lück, Wolfgang; Weinberger, Shmuel On the stable Cannon conjecture. J. Topol. 12 (2019), no. 3, 799–832.