Bounds on the genus of Riemann surfaces under certain group actions

Abstract

The symmetric genus σ (G) of a finite group G is the smallest genus of those compact Riemann surfaces on which G acts faithfully, possibly reversing orientation. A natural question to ask is whether there exists a finite group G with σ (G) = g for any given non-negative integer g. It is known that the values of the function σ cover well over 88% of all non-negative integers, and it has been conjectured that any remaining gaps in the range of the function σ can be filled using metabelian groups. In this thesis, we determine the symmetric genus of several infinite families of non-abelian and non-dihedral metacyclic groups, which are a subset of metabelian groups, including the families Cm:C2, Cm : Cp with p an odd prime, and Cm : C4. We find that the families Cm : Cp help fill in some gaps, leaving only five candidates for possible gaps in the symmetric genus range less than 1000. We also study the related topic of group actions on pseudo-real surfaces. A compact Riemann surface is called pseudo-real if it admits anti-conformal automorphisms, but no anti-conformal automorphism of order 2. It is known that there exist pseudo-real surfaces of every possible genus g ≥ 2. In this thesis, we extend the concept of the symmetric genus by defining two new parameters: the pseudo-real genus ψ (G) of a finite group G is the smallest genus of those pseudo-real surfaces on which G acts faithfully, possibly preserving orientation, while the strong pseudo-real genus ψ* (G) of G is the smallest genus of those pseudo-real surfaces on which G acts faithfully, without preserving orientation, when this exists. Our main result is that for every integer g ≥ 2, there exists a finite group G for which ψ (G) = ψ* (G) = g, and hence there are no gaps in the range of each of the functions ψ and ψ*. We also give an example of a group G for which ψ* (G) is defined but ψ (G) < ψ* (G).