A Moduli Space of Marked Hyperbolic Structures for Big Surfaces

We introduce the moduli space of marked, complete, Nielsen-convex hyperbolic structures on a surface of negative, but not necessarily finite, Euler characteristic. The emphasis is on infinite type surfaces, the aim being to study mapping class groups of infinite type surfaces via their action on this marked moduli space. We define a topology on the marked moduli space and prove that it reduces to the usual Teichm\"uller space for finite type surfaces. We also prove that the action of the mapping class group on this marked moduli space is continuous.