2021/11/30 by Arnab Kundu, Kundu, Arnab, Dipendu Maity +1
Mathematics · #Finite Group Theory Research #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2111.15484
If the face-cycles at all the vertices in a map are of same type then the map is called semi-equivelar. In particular, it is called equivelar if the face-cycles contain same type of faces. A map is semiregular (or almost regular) if it has as few flag orbits as possible for its type. A map is k-regular if it is equivelar and the number of flag orbits of the map k under the automorphism group. In particular, if k =1, its called regular. A map is k-semiregular if it contains more number of flags as compared to its type with the number of flags orbits k. Drach et al. \citedrach:2019 have proved that every semi-equivelar toroidal map has a finite unique minimal semiregular cover. In this article, we show the bounds of flag orbits of semi-equivelar toroidal maps, i.e., there exists k for each type such that every semi-equivelar map is ℓ-uniform for some ℓ ≤ k. We show that none of the Archimedean types on the torus is semiregular, i.e., for each type, there exists a map whose number of flag orbits is more than its type. We also prove that if a semi-equivelar map is m-semiregular then it has a finite index t-semiregular minimal cover for t ≤ m. We also show the existence and classification of n sheeted k-semiregular maps for some k of semi-equivelar toroidal maps for each n ∈ ℕ.