Construction of the discrete hull for the combinatorics of a regular pentagonal tiling of the plane

The article A “regular” pentagonal tiling of the plane by P. L. Bowers and K. Stephenson, Conform.Geom. Dyn. 1, 58–86, 1997, defines a conformal pentagonal tiling. This is a tiling of the planewith remarkable combinatorial and geometric properties. However, it doesn’t have finite localcomplexity in any usual sense, and therefore we cannot study it with the usual tiling theory. Theappeal of the tiling is that all the tiles are conformally regular pentagons. But conformal mapsare not allowable under finite local complexity. On the other hand, the tiling can be describedcompletely by its combinatorial data, which rather automatically has finite local complexity. Inthis paper we give a construction of the discrete hull just from the combinatorial data. The mainresult of this paper is that the discrete hull is a Cantor space.