Algebraic groups whose orbit closures contain only finitely many orbits

TL;DR

This paper classifies connected affine algebraic groups with the property that all orbit closures contain finitely many orbits, showing they are either tori or products of tori with one-dimensional unipotent groups.

Contribution

It provides a complete classification of such groups and characterizes them via the modality of their actions, linking geometric and algebraic properties.

Findings

Groups with property (F) are tori or products of tori and unipotent groups.

Property (F) is equivalent to a specific modality condition.

The classification connects orbit closure finiteness with group structure.

Abstract

We explore connected affine algebraic groups $G$, which enjoy the following finiteness property $\rm (F)$: for every algebraic action of $G$, the closure of every $G$-orbit contains only finitely many $G$-orbits. We obtain two main results. First, we classify such groups. Namely, we prove that a connected affine algebraic group $G$ enjoys property $\rm (F)$ if and only if $G$ is either a torus or a product of a torus and a one-dimensional connected unipotent algebraic group. Secondly, we obtain a characterization of such groups in terms of the modality of action in the sense of V. Arnol'd. Namely, we prove that a connected affine algebraic group $G$ enjoys property $\rm (F)$ if and only if for every irreducible algebraic variety $X$ endowed with an algebraic action of $G$, the modality of $X$ is equal to $\dim X-\max_{x\in X} G\cdot x$.