Finite dimensional groups of local diffeomorphisms

TL;DR

This paper classifies certain groups of local biholomorphisms as algebraic groups, especially virtually polycyclic and finitely generated virtually nilpotent groups, and extends uniform intersection multiplicity estimates to these classes.

Contribution

It establishes that virtually polycyclic groups of local biholomorphisms can be given a canonical algebraic group structure, generalizing previous results.

Findings

Virtually polycyclic groups can be endowed with a canonical algebraic structure.

Methods to identify algebraic group structures in groups of local diffeomorphisms.

Extension of uniform intersection multiplicity estimates to broader classes of groups.

Abstract

We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic group up to add extra formal diffeomorphisms. We show that this is the case for virtually polycyclic subgroups and in particular finitely generated virtually nilpotent groups of local biholomorphisms. We provide several methods to identify this property and build examples. As a consequence we generalize results of Arnold, Seigal-Yakovenko and Binyamini on uniform estimates of local intersection multiplicities to bigger classes of groups, including for example virtually polycyclic groups.