  We say that a group $G$ of local (maybe formal) biholomorphisms satisfies the
uniform intersection property if the intersection multiplicity $(\phi (V), W)$
takes only finitely many values as a function of $G$ for any choice of analytic
sets $V$ and $W$. In dimension $2$ we show that $G$ satisfies the uniform
intersection property if and only if it is finitely determined, i.e. there
exists a natural number $k$ such that different elements of $G$ have different
Taylor expansions of degree $k$ at the origin. We also prove that $G$ is
finitely determined if and only if the action of $G$ on the space of germs of
analytic curves have discrete orbits.
