A hierarchical strongly aperiodic set of tiles in the hyperbolic plane

We give a new construction of strongly aperiodic set of tiles in H, exhibiting a kind of hierarchical structure, simplifying the central framework of Margenstern’s proof that the Domino problem is undecidable in the hyperbolic plane [13]. Ludwig Danzer once asked whether, in the hyperbolic plane, where there are no similarities, there could be any notion of hierarchical tiling—an idea which plays a great role in many constructions of aperiodic sets of tiles in the Euclidean plane [1, 2, 4, 5, 6, 15, 17, 18]. It is an honor to dedicate this paper, which exposes a way to look at this question, to Herr Prof. Danzer in his 80th year. In 1966, R. Berger proved that the Domino Problem— whether a given set of tiles admits a tiling— is undecidable in the Euclidean plane, hanging his proof on the construction of an aperiodic set of tiles [2]. This first set was quite complex, with over 20,000 tiles; Berger himself reduced this to 104 [3] and in 1971, R. Robinson streamlined Berger’s proof of the undecidability of the Domino Problem, working off of an aperiodic set of just six tiles [18]. Both Berger’s and Robinson’s constructions used, in a very strong way, the hierarchical nature of their underlying aperiodic sets of tiles. In 1977, Robinson considered, but was unable to settle, the undecidability of the Domino Problem in the hyperbolic plane [19]. M. Margenstern recently gave a proof that the Domino Problem is undecidable in the hyperbolic plane [13]; despite the lack of scale invariance in this setting, he found a way to adapt and extend the Berger-Robinson construction. (J. Kari has independently given a completely different and highly original proof [11].) Though it is difficult to discern—and Margenstern does not mention—the more than 18,000 tiles underlying his construction are a strongly aperiodic set Over time, it became clear that when considering tilings outside of the Euclidean plane (in higher dimensions, or in curved spaces) one might distinguish between weakly aperiodic and strongly aperiodic sets of tiles [16]. Weakly aperiodic sets of tiles admit only tilings without a co-compact symmetry, i.e. without a compact fundamental domain. In the hyperbolic plane, this is an almost trivial property, enjoyed, for example, by the tiles in the n-fold horocyclic tiling described below. Strongly aperiodic sets of tiles, in contrast, admit only tilings with no period whatsoever, tilings on which there is no infinite cyclic action. In the Euclidean plane, the two properties