Boundness of intersection numbers for actions by two-dimensional biholomorphisms

TL;DR

This paper characterizes when groups of local biholomorphisms in two dimensions have finitely many intersection multiplicities, linking this property to finite determinacy and discrete orbit actions on germs of curves.

Contribution

It establishes an equivalence between the uniform intersection property, finite determinacy, and discrete orbit actions for groups of 2D biholomorphisms.

Findings

Groups with the uniform intersection property are finitely determined.

Finitely determined groups have discrete orbits on germs of analytic curves.

The paper provides a complete characterization of these groups in dimension two.

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.