Unboundedness of the Heesch Number for Hyperbolic Convex Monotiles

We provide a resolution of the Heesch problem for homogeneous (also known as semi-regular) tilings, and as a corollary, for tilings by convex monotiles in the hyperbolic plane. We also provide the first known example of weakly aperiodic convex monotiles arising from the dual of homogeneous tilings.