Here are two questions about finitely generated and finitely presented groups (FP):
- Is there an example of an FP group that does not admit a homomorphism to $\operatorname{GL}(n,C)$ with trivial kernel for any n?
The second question is modified according to the sujestion of Greg below.
- For which $n$ given two subgroups of $\operatorname{GL}(n,C)$ generated by explicit lists of matrices, together with finite lists of relations and the promise that they are sufficient, is there an algorithm to determine if they are isomorphic as groups?"
In both cases we don't impose any conidtion on the group (apart from been FP), in particular it need not be discrete in $\operatorname{GL}(n,C)$.
Best Answer
Here is a more complete picture to go with David's and Richard's answers.
It is a theorem of Malcev that a finitely presented group $G$ is residually linear if and only if it is residually finite. The proof is very intuitive: The equations for a matrix representation of $G$ are algebraic, so there is an algebraic solution if there is any solution. Then you can reduce the field of the solution to a finite field, as long as you avoid all primes that occur in the denominators of the matrices.
The same proof shows that $G$ has no non-trivial linear representations if and only if it has no subgroups of finite index. So Higman's group has this property.
A refined question is to find a finitely presented group which is residually finite, but nonetheless isn't "linear" in the sense of having a single faithful finite-dimensional representation. It seems that the automorphism group of a finitely generated free group, $\text{Aut}(F_n)$, is an example. Nielsen found a finite presentation for this group, it is also known to be residually finite, yet Formanek and Procesi showed that it is not linear when $n \ge 3$. More recently, Drutu and Sapir found an example with two generators and one relator.