Firkin, Adam (2015)
Ph.D. thesis, University of Birmingham.
Firkin15PhD.pdf
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 AbstractFor 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 onetoone 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 \(S\)\(_4\).

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