# Connectivity of Hurwitz spaces

James, Adam (2013). Connectivity of Hurwitz spaces. University of Birmingham. Ph.D.

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## Abstract

Let G be a finite group and C = (C1; : : : ;Cr) a collection of conjugacy classes of G. The Hurwitz space H(G;C) is the space of Galois covers of the Riemann Sphere with monodromy group G, and ramification type C. Points of the Hurwitz space can be parameterised combinatorially by Nielsen tuples: tuples of elements of G with product one. There is a correspondence between connected components of H(G;C) and orbits of the braid group on the set of Nielsen tuples.

In this thesis we consider the problem of determining the number of components of the Hurwitz space for A$$_5$$ and A$$_6$$. For both groups we give a complete classification of the braid orbits for all types C. We show that when there exists more than one orbit then Fried's lifting invariant distinguishes these orbits.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Shpectorov Prof., SergeyUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Mathematics
Funders: Engineering and Physical Sciences Research Council
Subjects: Q Science > QA Mathematics
URI: http://etheses.bham.ac.uk/id/eprint/4389

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