  Suppose that a group $G$ acts non-elementarily on a hyperbolic space $S$ and
does not fix any point of $\partial S$. A subgroup $H\le G$ is said to be
geometrically dense in $G$ if the limit sets of $H$ and $G$ coincide and $H$
does not fix any point of $\partial S$. We prove that every invariant random
subgroup of $G$ is either geometrically dense or contained in the elliptic
radical (i.e., the maximal normal elliptic subgroup of $G$). In particular,
every ergodic measure preserving action of an acylindrically hyperbolic group
on a Borel probability space $(X,\mu)$ either has finite stabilizers
$\mu$-almost surely or otherwise the stabilizers are very large (in particular,
acylindrically hyperbolic) $\mu$-almost surely.
