Finite groups of small genus

Mohammed Salih, Haval M. (2015). Finite groups of small genus. University of Birmingham. Ph.D.

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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.