20.46 (2022)

A group action on a compact space is said to be topologically free if the set of points with trivial stabilizer is dense. Let $G$ be a locally compact group, and $\partial_{\text{sp}} G$ its Furstenberg boundary (the largest minimal and strongly proximal compact $G$-space). Let $\Gamma_1$ and $\Gamma_2$ be two lattices in $G$, both acting faithfully on $\partial_{\text{sp}} G$.
$\qquad$ a) Is it possible that the $\Gamma_1$-action on $\partial_{\text{sp}} G$ is topologically free, but the $\Gamma_2$-action on $\partial_{\text{sp}} G$ is not topologically free?
$\qquad$ b) If yes, can this also happen if $\partial_{\text{sp}} G = G/H$ is a homogeneous $G$-space?