Geodesic
Conjugacy classes of reflections of maps
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 70-99

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This paper considers how many conjugacy classes of reflections a map can have, under various transitivity conditions. It is shown that for vertex- and for face-transitive maps there is no restriction on their number or size, whereas edge-transitive maps can have at most four classes of reflections. Examples are constructed, using topology, covering spaces and group theory, to show that various distributions of reflections can be achieved. Connections with real forms of algebraic curves are also discussed. 
@article{ZNSL_2016_446_a4,
     author = {G. A. Jones},
     title = {Conjugacy classes of reflections of maps},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     volume = {446},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a4/}
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G. A. Jones. Conjugacy classes of reflections of maps. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 70-99. http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a4/