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

1) Is there an example of an FGFP FP group that does not admit a homomorphism to $GL(n,C)$ with trivial kernel for any n?

The second question is modified according to the sujestion of Greg below.

2) For which $n$ given two subgroups of $GL(n,C)$ 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 FGFP)FP), in particular it need not be discrete in $GL(n,C)$.
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Here are two questions about finitely generated and finitely presented groups (FGFP):

1) Is there an example of an FGFP group that does not admit a homomorphism to $GL(n,C)$ with trivial kernel?

The second question is modified according to the sujestion of Greg below.

2) For which $n$ given two subgroups of $GL(n,C)$ 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 FGFP), in particular it need not be a discreet discrete in $GL(n,C)$.
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Here are two questions about finitely generated and finitely presented groups (FGFP):

1) Is there an example of an FGFP group that does not admit a homomorphism to $GL(n,C)$ with trivial kernel?

The second question is modified according to the sujestion of Greg below.

2) For which $n$ it is possible to prove that FGFP given two subgroups of $GL(n,C)$ generated by explicit lists of matrices, together with finite lists of relations and the promise that they are algorithmically recognisable?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 FGFP), in particular it need not be a discreet in $GL(n,C)$.
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