Let $S$ be closed hyperbolic surface. The Dehn-Neilson theorem $\Gamma \approx Out(\pi_1)$ identifies the mapping class group of $S$ with the outer automorphism group of the surface group $\pi_1=\pi_1(S, pt)$.
Now $\pi_1$ has many faithful linear representations $\rho: \pi_1 \to GL_N$ into, say, large dimensional real or complex matrix groups $GL_N$. For a long time I've always wondered whether it was possible to construct a faithful linear representation $\rho$ of $\pi_1$ with the following property:
Construct a linear representation $\rho: \pi_1 \to GL_N$ such that the image $\rho':=im(\rho)=\rho(\pi_1)$ in $GL_N$ has a large normalizer group $N_{GL}(\rho')$ and the quotient $N_{GL}(\rho')/\rho'$ is finite index in $Out(\pi_1)$.
Evidently the normalizer is typically trivial and equal to $\rho'$. In effect, we are attempting to represent the outer automorphisms of $\pi_1$ by $G$-inner automorphisms, namely conjugation by matrices in the normalizer $N_{GL}(\rho')$. Is it possible that there are extreme special representations of $\pi_1$ with large normalizers?
I've had this idea for long time, and have tried to investigate it's plausibility, but I'm not sure if it's obviously impossible. Evidently a positive answer would imply that MCG is linear for closed surfaces. It was somewhat inspired by Skolem-Noether theorem, and makes me think the question can be phrased in terms of Serre's galois cohomology, but that is beyond my understanding.
Any references or insight most welcome!
