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

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. Shpectorov, Sergey (Prof.) Colleges (2008 onwards) > College of Engineering & Physical Sciences School of Mathematics QA Mathematics University of Birmingham 5702
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