Geodesic
Mirrors of reflections of regular maps
Ars Mathematica Contemporanea, Tome 15 (2018) no. 2, pp. 347-354.

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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.
DOI : 10.26493/1855-3974.1459.11d
Keywords: Riemann surface, regular map, Hurwitz map, reflection, mirror 
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Adnan Melekoğlu. Mirrors of reflections of regular maps. Ars Mathematica Contemporanea, Tome 15 (2018) no. 2, pp. 347-354. doi : 10.26493/1855-3974.1459.11d. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1459.11d/

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