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.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Magaard Dr, KayUNSPECIFIEDUNSPECIFIED
Shpectorov Prof., SergeyUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Mathematics
Funders: None/not applicable
Subjects: Q Science > QA Mathematics
URI: http://etheses.bham.ac.uk/id/eprint/5574

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