  In this paper, we develop the mathematical tools needed to explore isotopy
classes of tilings on hyperbolic surfaces of finite genus, possibly
nonorientable, with boundary, and punctured. More specifically, we generalize
results on Delaney-Dress combinatorial tiling theory using an extension of
mapping class groups to orbifolds, in turn using this to study tilings of
covering spaces of orbifolds. Moreover, we study finite subgroups of these
mapping class groups. Our results can be used to extend the Delaney-Dress
combinatorial encoding of a tiling to yield a finite symbol encoding the
complexity of an isotopy class of tilings. The results of this paper provide
the basis for a complete and unambiguous enumeration of isotopically distinct
tilings of hyperbolic surfaces.
