We compute the dimension of the tangent space to, and the Krull
dimension of, the prorepresentable hull of two deformation
functors. The first one is the "algebraic" deformation functor of an
ordinary curve $X$ over a field of positive characteristic with
prescribed action of a finite group $G$, and the data are computed in
terms of the ramification behaviour of $X\to G\backslash X$. The second
one is the "analytic" deformation functor of a fixed embedding of a
finitely generated discrete group $N$ in ${\rm PGL}(2, K)$ over a
nonarchimedean-valued field $K$, and the data are computed in terms of
the Bass-Serre representation of $N$ via a graph of groups. Finally,
if $\Gamma$ is a free subgroup of $N$ such that $N$ is contained in
the normalizer of $\Gamma$ in ${\rm PGL}(2, K)$, then the Mumford
curve associated to $\Gamma$ becomes equipped with an action of
$N/\Gamma$, and we show that the algebraic functor deforming the
latter action coincides with the analytic functor deforming the
embedding of $N$.