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.

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