On the existence of abelian groups of automorphisms of Klein surfaces

Computing groups of automorphisms of Riemann and Klein surfaces is a classical problem initiated by Schwartz, Hurwitz, Klein and Wiman, among others, at the end of the 19th century. Surfaces with a nontrivial finite group of automorphisms are of particular importance, since they correspond to the singular locus of the moduli space of such surfaces. By the uniformization theorem, compact Riemann and Klein surfaces of algebraic genus greater than one can be seen as the quotient of the hyperbolic plane under the action of a discrete subgroup of its isometries (a non-Euclidean crystallographic group, in general, or a Fuchsian group if it only contains orientation-preserving isometries). This approach gave rise to the use of combinatorial methods, which have proven the most fruitful in computing groups of automorphisms. Thus far, research has focused on low genus surfaces or on surfaces with a certain group of automorphisms endowing the surface with significant properties (for instance, hyperelliptic, elliptic-hyperelliptic, Wiman, Accola-Maclachlan and Kulkarni surfaces). Not surprisingly, cyclic groups were tackled firstly [39]. Combinatorial methods were first applied by Harvey [19]. He found necessary and sufficient conditions for a cyclic group to act on a Riemann surface. Such conditions are expressed in terms of the algebraic structure of the Fuchsian group associated to the action. Harvey’s Theorem has been widely used since. Similar results have only been found for dihedral [8] and abelian groups [3, Theorem 9.1]. For p-groups, the problem has been studied by Kulkarni and Maclachlan [21]. For cyclic actions on Klein surfaces with boundary, the result corresponding to Harvey’s Theorem was proven in [11, §3.1]. A similar theorem for abelian actions remained unknown, although some meaningfull, partial results were well-known, such as the answer to the minimum genus problem for cyclic [12] and noncyclic abelian groups [29]. Minimum genus and maximum order problems have been studied for a number of families of groups using diverse techniques. Some thorough surveys on these topics can be found in [9, 6, 7]. One of these techniques takes advantage of previously established conditions for the existence of surface-kernel epimorphisms onto a group of the family. This approach usually provides a shorter proof to the solution to the minimum genus and maximum order problems, as we will see in subsequent chapters. In this thesis, we obtain the following results: Chapter 2. We establish a refinement of Breuer’s conditions [3, Theorem 9.1] for the existence of abelian actions on compact Riemann surfaces of genus greater than one. In this new form, every condition is entirely expressed in terms of the invariant factors of the abelian group and the signature of the Fuchsian group. As a consequence, we obtain a new, shorter proof of Maclachlan’s solution to the minimum genus problem and, in many cases, an explicit expression using some results concerning the invariant factors of the abelian group. We find the least strong symmetric genus for the family of abelian groups, cyclic or not, of the same given order, as well as the unique abelian group attaining such minimum genus, which leads to a new proof of the maximum order problem for the family of abelian groups acting on Riemann surfaces of a given genus greater than one. These results were published in [32]. Chapter 3. We state conditions for an abelian group to act on some compact bordered Klein surfaces of algebraic genus greater than one, expressing such conditions in terms of the algebraic structure of the NEC group associated to that action. We then deduce by new, more concise methods the real genus of an abelian group and solve the related maximum order problem. We also find the expression for the least real genus of abelian groups of the same given order. The results in this chapter are already published in [33]. Chapter 4. We find conditions of existence of actions of abelian groups of odd order or with cyclic Sylow 2-subgroup on compact nonorientable Riemann surfaces of topological genus greater than two. That makes it easier to obtain the known expression of the symmetric cross-cap number of such groups.