Mirrors of reflections of regular maps

Adnan Melekoğlu

Abstract


A regular map ℳ is an embedding of a finite connected graph into a compact surface S such that its automorphism group Aut+(ℳ) acts transitively on the directed edges. A reflection of ℳ fixes a number of simple closed geodesics on S, which are called mirrors. In this paper, we prove two theorems which enable us to calculate the total number of mirrors fixed by the reflections of a regular map and the lengths of these mirrors. Furthermore, by applying these theorems to Hurwitz maps, we obtain some interesting results. In particular, we find an upper bound for the number of mirrors on Hurwitz surfaces.


Keywords


Riemann surface, regular map, Hurwitz map, reflection, mirror

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DOI: https://doi.org/10.26493/1855-3974.1459.11d

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications