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

Firkin, Adam (2015)
Ph.D. thesis, University of Birmingham.

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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 \(S\)\(_4\).

Type of Work:Ph.D. thesis.
Supervisor(s):Shpectorov, Sergey (Prof.)
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Mathematics
Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:5702
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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