Conjugacy classes of reflections of maps Seminar

A compact Riemann surface S can be regarded as a complex algebraic curve C, and by Belyi's Theorem this is defined over an algebraic number field if and only if it is obtained from a map (= dessin d'enfant = graph embedding); the conjugacy classes of orientation-reversing involutions of S then correspond to the real forms of C, and the reflections correspond to those with real points.

This talk 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. There are connections with work by Natanzon, by Bujalance, Gromadzki and Izquierdo, and by Bujalance and Singerman on symmetries of Riemann surfaces, and by Melekoglu and Singerman on patterns of reflections of regular maps.