Symmetries and automorphisms of compact Riemann surfaces

Abstract

In this thesis we deal with compact Riemann surfaces, in fact mainly those uni- formized by normal subgroups of Fuchsian triangle groups. A symmetry of such a surface is an anti-conformal involution mapping the surface to itself. Each sym- metry is given a species which completely classifies its topological action on the surface. We examine an oversight in an important Theorem of Singerman's and try to mend it. In so doing we find a kind of symmetry not anticipated by Singer- man. Chapter four contains the symmetry types of all Riemann surfaces with large cyclic automorphism groups. Harnack gave an upper bound on the number of curves fixed by a symmetry of a surface of a particular genus, the 'unexpected' symmetries mentioned above are the only symmetries of the surfaces in Chapter four to attain that bound. In Chapter five we give a similar treatment to those surfaces with large non-cyclic abelian automorphism groups. Harnack's bound is not attained by any symmetry of any of these surfaces. The Appendix chiefly accompanies Chapters four and five and looks in some detail at the inclusions between triangle groups and the NEC groups that contain them with index two. In Chapter six we turn our attention to maps and hypermaps, lying on ori- entable, connected surfaces without boundary. Such objects can naturally be thought of as lying on those Riemann surfaces in the scope of this thesis. These surfaces, together with the maps and hypermaps themselves, are receiving much attention at the moment in connection with Belyi's Theorem, which implies that they are precisely the surfaces corresponding to the algebraic curves defined over algebraic number fields. The maps and hypermaps that we deal with are all regu- lar and their face centres, vertices and edge centres are important in the geometric and combinatorial properties of the maps and hypermaps. We call these points 'geometric points'. Weierstrass points are right at the heart of Riemann surfaces and we determine whether the 'geometric' points of regular maps and hypermaps with abelian automorphism groups are Weierstrass points or not. Finally we cal- culate the weight at each of the 'geometric' points of all the regular maps of genus two, three, four and five.

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