Symmetries of Riemann surfaces and regular maps

In this thesis we concentrate on symmetric Riemann surfaces. By a symmetric surface we mean a surface admitting an anti-conformal involution which we call a 'symmetry'. The fixed point set of a symmetry of a Riemann surface consists of disjoint simple closed curves which are called 'mirrors'.In chapter one we give the background material and in chapter two we introduce Hoare's Theorem which we use to find the symmetry type of a Riemann surface. In chapter three we find the symmetry types of Riemann surfaces of genus three.Harnack proved that a symmetry of a Riemann surface of genus g cannot have more than g + 1 mirrors. When this bound is attained the corresponding surface is called the 'M-surface' and the symmetry with g + 1 mirrors is called an 'M-symmetry'. Similarly, a symmetry with g mirrors is called an '(M-1)-symmetry' and the corresponding surface is called an '(M-1)-surface'. In chapter four we investigate those M-surfaces of genus g > 1 admitting an automorphism of order g + 1 which cyclically permutes the mirrors of an M-symmetry. We call such a surface an M-surface with the 'M-property'. We also show that for every g > 1, there is a unique Platonic M and M-1 surface of genus g. By a Platonic surface we mean a surface uniformised by a normal subgroup of a Fuchsian group with signature [2, m, n].In chapter five we look at reflexible regular maps on surfaces. A reflection of such a map is a symmetry of the underlying surface and its mirrors always pass through some 'geometric points' (vertices, edge-centres and face-centres) of the map. The geometric points lying on every mirror form a sequence which we call the 'pattern' of the mirror. In chapter five we investigate the patterns of mirrors on Riemann surfaces of genus g > 1 and show that there are Platonic Riemann surfaces of genus g > 3 which admit only one conjugacy class of reflective symmetries.