The topology of the minimal regular covers of the Archimedean tessellations
A map \(M=(\Gamma,i,S)\) is a 2-cell embedding \(i\) of a locally finite graph \(\Gamma\) on a surface \(S\). \textit{M. I. Hartley} [Discrete Comput. Geom. 21, No. 2, 289--298 (1999; Zbl 0927.52017)] provided a procedure to describe an arbitrary map \(M\) algebraically in terms of a regular cover \(\widetilde{M}=(\widetilde{\Gamma}, \widetilde{i},\widetilde{S})\) and subgroup of its authomorphisms. In the reviewed paper it is proved that if \(M=(\Gamma,i,\mathbb{E}^2)\) is a periodic map in Euclidian space with convex faces such that it contains at least two faces with different number of edges, and \(\widetilde{M}=(\widetilde{\Gamma}, \widetilde{i},\widetilde{S})\) is its minimal regular cover, then \(\widetilde{S}\) is homeomorphic to the topological surface obtained by gluing infnitely many torii along a ray, known as Loch Ness Monster.
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