Compact hyperbolic polygons tiling right-angled polygons

Classify all compact hyperbolic polygons that tile a right-angled hyperbolic polygon.

Background

The paper develops a three-dimensional classification using a two-dimensional auxiliary problem concerning Coxeter polygons that admit ideal polygons tiled by their reflection-group images. A theorem of Felikson supplies a classification of the idealizable Coxeter polygons relevant to the paper’s arguments.

The authors identify a related two-dimensional problem that remains unresolved: a complete classification when the tile is compact and the target polygon is right-angled.