Finite groups and coverings of surfaces

Kazaz, Mustafa (1997) Finite groups and coverings of surfaces. University of Southampton, Doctoral Thesis.

Record type: Thesis (Doctoral)

Abstract

In this thesis the theory of regular coverings of compact, connected surfaces with a given finite covering group G is investigated, in particular finite abelian coverings of the regular hypermaps of genus 2 and their reflexibilities are determined. The aim is to apply techniques from finite group theory, such as character theory and Philip Hall's algebraic version of the Möbius Inversion Formula, to study coverings of surfaces. The first two chapters review well-known facts about group theory, number theory, P. Hall's Möbius Inversion Formulae, general covering spaces, and maps and hypermaps on compact orientable surfaces without boundary. In Chapter 2, after outlining the basic proprieties of representations and characters of finite groups, we discuss how character theory can be used to count solutions of equations in finite groups. In Chapter 3 we combine the character-theoretic techniques and P. Hall's method to obtain the number of equivalence classes of regular coverings of a compact, connected surface with a given finite covering group G. Chapter 4 is concerned with the action of Aut II_g on the set N_g(G) = {N ≤ Π_g | Π_g / N symbol_187 G} of normal subgroups of the surface group Π_g, or equivalently how the self-homomorphisms of Σ_g permute its regular unbranched converings.

Finally, in Chapters 5 and 6 we turn our attention to regular hypermaps of genus 2. In Chapter 5 we find the homology representations and characters of the orientation-preserving automorphism groups of genus 2 regular hypermaps. In Chapter 6, following a method of Macbeath, we use these representations and characters to obtain finite abelian coverings of these hypermaps H of type (l, m, n) corresponding to hypermap-subgroups N ≤ Δ = Δ(l, m, n), and to study the reflexibility of these coverings: any G-submodule of the first homology group with coefficients in the integers mod q, H₁(S, Z_q) symbol_187 N / N' N^q symbol_187 Z⁴_q (q prime), corresponds to a normal subgroup M ≤ Δ lying between N and N' N^q. Thus we obtain a regular covering of H, with the finite abelian group N/M as the group of covering transformations.