On super-rigidity of Gromov's random monster group

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 ${\mathrm{\Gamma }}_{\alpha }$ 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 ${\mathrm{\Gamma }}_{\alpha }$ 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 ${\mathrm{\Gamma }}_{\alpha }$ has super-rigidity and hereditary super-rigidity.