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

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

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 $SL_n({\mathbb F}_p[[t]])$,...
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