On Super-rigidity of Gromov's Random Monster Group

Abstract In this article, we show super-rigidity of Gromov’s random monster group. It is known from a paper of Assaf Naor and Lior Silberman that any homomorphic image of Gromov’s random monster group into a linear group is finite. It can be also derived from the previously known results that the same result is true for a-Lp-menable groups and K-amenable groups. We extend these results and prove that any morphism ϕα from Gromov’s random monster group Γα to a countable discrete group G has finite image for almost all α, where G is any of the following types of groups: mapping class group MCG(Sg,b), braid group Bn, outer automorphism group of a free group Out(FN), automorphism group of a free group Aut(FN) and hierarchically hyperbolic group. For acylindrically hyperbolic groups, we deduce that the homomorphic image is absolutely elliptic. We introduce another property called hereditary super-rigidity. It immediately follows from the literature that Γα has hereditary super-rigidity with respect to an a-Lp-menable group or a K-amenable group for a.e. α. In this article, we establish a stability result for the groups with respect to which Γα has super-rigidity and hereditary super-rigidity. Mathematics Subject Classification (2020): 20F65, 20F67, 20P05, 53C24, 57K20.