If a translation by a group element on $G/\Gamma$ is minimal, is that element nilpotent in $\mathfrak G ?$ (i.e. has 0 entropy)
The more commonly used term here instead of nilpotent is "quasi-unipotent". By using ideas related to Schmidt games, in the joint paper with Dmitry Kleinbock, "Modified Schmidt games and a conjecture of Margulis," J. Mod. Dyn. 7 (2013) we show that elements which are not quasiunipotent have orbits which miss open sets and thus cannot act minimally. I imagine there are simpler proofs in the literature, e.g. in earlier papers of Starkov and Kleinbock-Margulis.