Random quotients of mapping class groups are quasi-isometrically rigid

In this paper we prove several rigidity properties for random quotients of mapping class groups of finite-type surfaces, namely whose kernel is normally generated by random walks. Firstly, every automorphism of the corresponding quotient of the curve graph is induced by a mapping class, thus generalising a celebrated theorem of Ivanov's. Next, we show that, if a finitely generated group is quasi-isometric to one such random quotient, then the two groups are weakly commensurable. This uses techniques from the world of hierarchically hyperbolic groups, and indeed in the process we clarify a proof of Behrstock, Hagen, and Sisto on the quasi-isometric rigidity of mapping class groups. Finally, we show that the automorphisms groups of our quotients, as well as their abstract commensurators, coincide with the groups themselves. Our results hold for a wider family of quotients, roughly those whose kernel act by sufficiently large translations on the curve graph. This class also includes quotients by suitable powers of a pseudo-Anosov element.