Abstract: For a compact surface S, $MCG(S) = Homeo(S)/Homeo_0(S)$ is a group well-studied and loved by many, but especially by geometric group theorists because it is finitely generated, and hence has a coarse geometry coming from its Cayley graph. When S is big (infinite-type), $MCG(S)$ is Polish and no longer discrete, but following the work of Rosendal and Mann-Rafi, we can still define a coarse geometry and a Cayley graph. In ongoing work with Schaffer-Cohen—Verberne and Qing—Rafi, we can show that for some surfaces, $MCG(S)$ is non-elementary $\delta$-hyperbolic, and some have infinite coarse rank. If you like surfaces, geometry, topology, and Polish groups, come by!