Abstract Commensurations of Big Mapping Class Groups
It is a classic result of Ivanov that the mapping class group of a finite-type surface is equal to its own automorphism group. Relatedly, it is well-known that non-homeomorphic surfaces cannot have isomorphic mapping class groups. In the setting of “big mapping class groups” of infinite-type surfaces, the situation is more complicated due to the fact that the sheer enormity and variety of behavior prevents group elements from having canonical descriptions in terms of normal forms. This talk will present work with Juliette Bavard and Kasra Rafi overcoming these difficulties and extending the above results to big mapping class groups. In particular, we show that any isomorphism between big mapping class groups is induced by a homeomorphism of the surfaces and that each big mapping class group is equal to its abstract commensurator.