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We consider quotients of mapping class groups of orientable, finite typesurfaces by subgroups whose action on the curve graph has large displacement.This class includes quotients by the normal closure of a pseudo-Anosov element,the mapping class group itself, and in view of forthcoming work of Abbott,Berlyne, Ng, and Rasmussen, also random quotients. First, we show that everyautomorphism of the corresponding quotient of the curve graph is induced by amapping class, thus generalising Ivanov's Theorem about automorphisms of thecurve graph. Then we use this to prove quasi-isometric rigidity underadditional assumptions, satisfied by all aforementioned quotients. In theprocess, we clarify a proof of quasi-isometric rigidity of mapping class groupsby Behrstock, Hagen, and Sisto. Finally, we show that the outer automorphismsgroups of our quotients, as well as their abstract commensurators, are "thesmallest possible".

Rigidity Results for large displacement quotients of mapping class groups