What are all the possible geometric shapes of topological surfaces? Can one make a space out of them? The answer is yes! In this talk, I will talk about how to make a space out of all the (specific type of) possible hyperbolic structures on a surface, which I call the marked moduli space. I will also talk about how to construct a hyperbolic structure on the ladder surface which admits an infinite order isometry (of the hyperbolic surface). This isometry plays a key role in proving that there is no MCG-invariant metric on the marked moduli space of the ladder surface.