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

Boundness of intersection numbers for actions by two-dimensional biholomorphisms