Boundness of intersection numbers for actions by two-dimensional biholomorphisms

Abstract: 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.