Mapping class groups of finite-type surfaces have often been studied via their action on various graphs, whose vertices correspond to homotopy classes of certain arcs or curves on the surface, with edges between homotopically disjoint arcs or curves. In 2006, Ivanov posed a ​"metaconjecture": every object naturally associated to a surface, which admits an action of the mapping class group, with sufficiently rich structure has its group of automorphisms be the mapping class group. This is true for a lot of these arc/curve graphs — in particular, this means that the combinatorial structure of these graphs perfectly captures the topology of the surface.

In this talk, we shall explore Ivanov’s metaconjecture for mapping class groups of infinite-type surfaces. We shall focus on the action on “the grand arc graph”, first introduced by Bar-Natan—Verberne, which is an infinite-diameter, hyperbolic graph defined on a large class of infinite-type surfaces. This talk contains joint work with G. Shaji.