  A. Vistoli observed that, if Grothendieck's section conjecture is true and
$X$ is a smooth hyperbolic curve over a field finitely generated over
$\mathbb{Q}$, then $\underline{\pi}_{1}(X)$ should somehow have essential
dimension $1$. We prove that an infinite, pro-finite \'etale group scheme
always has infinite essential dimension. We introduce a variant of essential
dimension, the fce dimension $\operatorname{fced} G$ of a pro-finite group
scheme $G$, which naturally coincides with $\operatorname{ed} G$ if $G$ is
finite but has a better behaviour in the pro-finite case. Grothendieck's
section conjecture implies $\operatorname{fced}\underline{\pi}_{1}(X)=\dim X=1$
for $X$ as above. We prove that, if $A$ is an abelian variety over a field
finitely generated over $\mathbb{Q}$, then
$\operatorname{fced}\underline{\pi}_{1}(A)=\operatorname{fced} TA=\dim A$.
