Here are two questions about finitely generated and finitely presented groups (FP):


1. 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 [suggestion](https://mathoverflow.net/a/9635) of Greg below.


2. 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 being FP), in particular it need not be discrete in $\operatorname{GL}(n,C)$.



Finitely presented sub-groups of $\operatorname{GL}(n,C)$
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](https://doi.org/10.1112/jlms/s2-10.3.265 "McCool, J. On Nielsen’s presentation of the automorphism group of a free group. J. Lond. Math. Soc., II. Ser. 10, 265-270 (1975). zbMATH review at https://zbmath.org/0338.20029") found a finite presentation for this group, it is also [known to be residually finite](https://web.archive.org/web/20100629064551/http://www.math.lsu.edu/%7Ekasten/Papers/abs_lerf.html "Dasbach, Oliver T.; Mangum, Brian S. The automorphism group of a free group is not subgroup separable. Gilman, Jane (ed.) et al., Knots, braids, and mapping class groups. AMS/IP Stud. Adv. Math. 24, 23-27 (2001). zbMATH review at https://zbmath.org/1006.20027"), yet [Formanek and Procesi](https://doi.org/10.1016/0021-8693(92)90029-L "Formanek, Edward; Procesi, Claudio. The automorphism group of a free group is not linear. J. Algebra 149, No. 2, 494-499 (1992). zbMATH review at https://zbmath.org/0780.20023") showed that it is not linear when $n \ge 3$. More recently, [Druţu and Sapir](https://arxiv.org/abs/math/0405470 "Druţu, Cornelia; Sapir, Mark. Non-linear residually finite groups. J. Algebra 284, No. 1, 174-178 (2005), doi:10.1016/j.jalgebra.2004.06.025. zbMATH review at https://zbmath.org/1076.20021") found an example with two generators and one relator.


