An infinite profinite group such that any $p$-adic representation has finite image

Question

Fix a prime $ p $. We call an infinite profinite group $ G $ a Fontaine-Mazur group (with respect to $ p $) if every continuous homomorphism $ G\to {\rm GL}_n(\overline{\mathbb{Q}}_p) $ has finite image for any positive integer $ n $. The motivation for defining such a group comes from the following example:

Let $ K $ be a number field, $ S $ a finite set of primes of $ K $ not containing any primes above $ p $ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ unramified outside $ S $. Then the unramified Fontaine-Mazur Conjecture claims that $ G_{K,S} $ is a Fontaine-Mazur group (with respect to $ p $), cf. Conjecture 5a in [Fontaine and Mazur. "Geometric Galois representations." Elliptic curves, modular forms, Fermat’s last theorem (Hong Kong, 1993).]

My question is that: can someone give more examples of Fontaine-Mazur groups? For example, is $ {\rm SL}_n(\mathbb{F}_p[[T]]) $ a Fontaine-Mazur group? More generally, for any infinite complete Noetherian local $ \mathbb{F}_p $-algebra $ A $, is $ {\rm SL}_n(A) $ a Fontaine-Mazur group?

Answer 1

The group $SL_n({\mathbb F}_p[t])$, for $n \geq 3$ has super rigidity property and hence any representation over characteristic zero has finite image. Since this is dense in $SL_n({\mathbb F}_p[[t]])$, the latter group has the property that its representations over char zero have finite image

Answer 2

Let $G$ be a finitely generated, residually finite group for which every linear representation in char. zero has a finite image. For instance, this holds if $G$ is a torsion group. An example of such an infinite group is Grigorchuk's group.

Let $H$ be the profinite completion of $G$ (so $H$ is infinite as soon as $G$ is infinite). Then $H$ has the required property for every prime $p$ (just because it has a dense copy of $G$, which makes the job). (Actually all that matters is that $G\to H$ has a dense image, i.e. $H$ is a quotient of the profinite completion.)

[Note: Actually the examples provided by Venkataramana are of this form. They also cover $n=2$, for which one has to rather pick $\mathrm{SL}_2(\mathbf{F}_p[t,(1+t)^{-1}])$ to apply superrigidity. ]