Let $G$ be a connected real semi-simple Lie group and $H$ a closed connected subgroup. Let $P$ be a minimal parabolic subgroup of $G$. It is shown that $H$ has an open orbit on the flag manifold $G/P$ if and only if it has finitely many orbits on $G/P$. This confirms a conjecture by T. Matsuki.

We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between $S\left(G\right)$, the Samuel compactification, and $E\left(M\right(G\left)\right)$, the enveloping semigroup of the universal minimal flow. We resolve Furstenberg’s problem for several automorphism groups and give a detailed study in the case of $G=S_{\infty }$, leading us to define and investigate several new types...