  In this article, we show super-rigidity of Gromov's random monster group. We
prove that any morphism $\phi_\alpha$ from Gromov's random monster group
$\Gamma_\alpha$ to the group $G$ has finite image for almost all $\alpha$,
where $G$ is any of the following types of groups: mapping class group
$MCG(S_{g,b})$, braid group $B_n$, outer automorphism group of a free group
$Out(F_N)$, automorphism group of a free group $Aut(F_N)$, hierarchically
hyperbolic group, a-$L^p$-menable group or K-amenable group. We introduce
another property called hereditary super-rigidity and prove that
$\Gamma_\alpha$ has hereditary super-rigidity with respect to an
a-$L^p$-menable group or a K-amenable group. We also establish a stability
theorem for the groups with respect to which $\Gamma_\alpha$ has super-rigidity
and hereditary super-rigidity.
