Algebraic groups whose orbit closures contain only finitely many orbits

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 $\dim X-\max_{x\in X} G\cdot x$.