  We study analytic deformations and unfoldings of holomorphic foliations in
complex projective plane $\mathbb{C}P(2)$. Let $\{\mathcal{F}_t\}_{t \in
\mathbb{D}_{\epsilon}}$ be topological trivial (in $\mathbb{C}^2$) analytic
deformation of a foliation $\mathcal{F}_0$ on $\mathbb{C}^2$. We show that
under some dynamical restriction on $\mathcal{F}_0$, we have two possibilities:
$\mathcal{F}_0$ is a Darboux (logarithmic) foliation, or $\{\mathcal{F}_t\}_{t
\in \mathbb{D}_{\epsilon}}$ is an unfolding. We obtain in this way a link
between the analytical classification of the unfolding and the one of its germs
at the singularities on the infinity line. Also we prove that a finitely
generated subgroup of $\mathrm{Diff}(\mathbb{C}^n,0)$ with polynomial growth is
solvable.
