Connectivity of Hurwitz spaces for $L$ $_2$ (7), $L$ $_2$ (11) and $S$ $_4$

Firkin, Adam (2015). Connectivity of Hurwitz spaces for $L$ $_2$ (7), $L$ $_2$ (11) and $S$ $_4$ . University of Birmingham. Ph.D.

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Abstract

For a finite group G and collection of conjugacy classes C = ( $C$ $_1$ ,…, $C$ $_r$ ). The (inner) Hurwitz space, H $^i$ $^n$ ( $G$ , C), is the space of Galois covers of the Riemann sphere with monodromy group isomorphic to $G$ and ramification type C. Such a space may be parameterized point wise by tuples, g = ( $g$ $_1$ ,…, $g$ $_r$ ) of $G$ , known as Nielsen tuples, such that $g$ $_1$ … $g$ $_r$ = 1 and $\langle$ $g$ $_1$ ,…, $g$ $_r$ $\rangle$ generate $G$ . The action of the braid group upon these Nielsen tuples is in a one-to-one correspondence with the connected components of Hurwitz spaces.

The aim of this thesis is to calculate the connected components of the Hurwitz space for the groups $L$ $_2$ (7), $L$ $_2$ (11) and $S$ $_4$ for any given type in the case of $L$ $_2$ ( $p$ ) and a particular class of types for $S$ $_4$ , using the method described. Furthermore, we establish that if two orbits exist we can distinguish these orbits via a lift invariant within the covering group $SL$ $_2$ (7) and $SL$ $_2$ (11) for $L$ $_2$ (7) and $L$ $_2$ (11) respectively, and any Schur cover for $S$ $_4$ .