Finite groups of small genus

Abstract

For a finite group $G$, the Hurwitz space $H$$^i$$_r$$_,$$^n$$_g$ ($G$) is the space of genus $g$ covers of the Riemann sphere with $r$ branch points and the monodromy group $G$. Let ε$_r$($G$) = {($x$$_1$,...,$x$$_r$) : $G$ = $\langle$$x$$_1$,...,$x$$_r$$\rangle$, Π$^r$$_i$$_=$$_1$ $x$$_i$ = 1, $x$$_i$ ϵ $G$#, $i$ = 1,...,$r$}. The connected components of $H$$^i$$_r$$_,$$^n$$_g$($G$) are in bijection with braid orbits on ε$_r$($G$). In this thesis we enumerate the connected components of $H$$^i$$_r$$_,$$^n$$_g$($G$) in the cases where $g$ $\leq$ 2 and $G$ is a primitive affine group. Our approach uses a combination of theoretical and computational tools. To handle the most computationally challenging cases we develop a new algorithm which we call the Projection-Fiber algorithm