Random quotients of mapping class groups are quasi-isometrically rigid

Abstract

We prove several rigidity properties for random quotients of mapping class groups of surfaces, namely whose kernel is normally generated by the n-th steps of finitely many independent random walks. Firstly, we generalise a celebrated theorem of Ivanov's: every automorphism of the corresponding quotient of the curve graph is induced by a mapping class. Next, we show that, if a finitely generated group is quasi-isometric to a random quotient, then the two groups are weakly commensurable. This uses techniques from the world of hierarchically hyperbolic groups: indeed, in the process we clarify a proof of Behrstock, Hagen, and Sisto on the quasi-isometric rigidity of mapping class groups, which might possibly be applied to other hierarchically hyperbolic 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, namely 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.