If I have a uniform hyperbolic tiling (not necessarily regular) and a polyomino (a connected set of cells of this tiling), how can I find out whether it's possible to cover the hyperbolic plane with the copies of this polyomino or not?
If the tiling is not regular, then relations between its polygons already constrain the possibilities a lot (for example, in the hyperbolic soccerball, (6,6,7), the ratio of hexagons to heptagons in any such polyomino must be precisely 7:3, and so only polyominoes with total number of cells divisible by 10 are acceptable). In the HyperRogue community, a few such shapes have been found, with 10, 40 or 50 cells and strong symmetry.
But is there some research into the general case?