On super-rigidity of Gromov's random monster group

In my talk, I will speak on the super-rigidity of Gromov's random monster group. It is a finitely generated random group $\Gamma_\alpha$ ( $\alpha$ is in a probability space $\mathcal{A}$) constructed using an expander graph by M. Gromov in 2000. It provides a counterexample to the Baum-Connes conjecture for groups with coefficients in commutative $C^*$-algebra. It is already known that it has global fixed point property for isometric affine action on $L^p$ spaces for $1< p<\infty$ (in particular, Property (T) ) for a.e. $\alpha$ due to Gromov and Naor-Silberman. It is also hyperbolically rigid, i.e., any isometric action of the group on a Gromov-hyperbolic space is elementary for a.e. $\alpha$ (due to Gruber-Sisto-Tessera). In this talk, I will discuss the following type of super-rigidity problem (motivated by Margulis super-rigidity theorem): for which countable group $G$, any collection of homomorphisms $\phi_\alpha:\Gamma_\alpha\rightarrow G$ have a finite image for a.e. $\alpha$? This question was first addressed for linear groups by Naor-Silberman. The super-rigidity follows immediately from the literature for groups with a-$L^p$-menablity and K-amenable groups. In this talk, we will show that $\Gamma_\alpha$ has super-rigidity with respect to the following groups $G$: mapping class group $MCG(S_{g,p})$, braid group $B_n$, automorphism group of a free group $Aut(F_n)$, outer automorphism group of a free group $Out(F_N)$ . We will also show a stability result of the class of groups $G$.