Algebraic Groups Whose Orbit Closures Contain Only Finitely Many Orbits

Abstract

We explore connected affine algebraic groups G, which enjoy the following finiteness property (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 (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 (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 − maxxϵX dim G∙x.