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## Algebraic groups whose orbit closures contain only finitely many orbits

arxiv.org. math. Cornell University, 2017. No. 1707.06914 [math.AG].
We classify all connected affine algebraic groups G such that there are only finitely many G-orbits in every algebraic G-variety containing a dense open G-orbit. We also prove that G enjoys this property if and only if every irreducible algebraic G-variety X is modality-regular, i.e., the modality of X (in the sense of V. Arnol’d) equals to that of a family which is open in X.