  Let $\Sigma_{g}$ be a closed, connected, and oriented surface of genus $g
\geq 24$ and let $\Gamma$ be a finite index subgroup of the mapping class group
$Mod(\Sigma_{g})$ that contains the Torelli group $\mathcal{I}(\Sigma_g)$. Then
any orientation preserving $C^1$ action of $\Gamma$ on the circle cannot be
faithful.
  We also show that if $\Gamma$ is a finite index subgroup of $Aut(F_n)$, when
$n \geq 8$, that contains the subgroup of IA-automorphisms, then any
orientation preserving $C^1$ action of $\Gamma$ on the circle cannot be
faithful.
  Similarly, if $\Gamma$ is a finite index subgroup of $Out(F_n)$, when $n \geq
8$, that contains the Torelli group $\mathcal{T}_n$, then any orientation
preserving $C^1$ action of $\Gamma$ on the circle cannot be faithful.
  In fact, when $n \geq 10$, any orientation preserving $C^1$ action of a
finite index subgroup of $Aut(F_n)$ or $Out(F_n)$ on the circle cannot be
faithful.
