[Madison, Wis.] : [University of Wisconsin--Madison], 2016.
1 online resource (vi, 63 pages) : illustrations
The monodromy of a family of varieties is a measure of how homology classes vary. Surprisingly, many familiar ideas in number theory, such as Galois representations and Cohen-Lenstra heuristics, are closely linked to the monodromy of specific families. In general, we expect the monodromy of a family to be ``big'', i.e. as large as possible subject to any geometrical or algebraic constraints arising from the family. In this thesis I study the monodromy of Hurwitz spaces of [G]-covers, moduli spaces for branched covers of the projective line with Galois group [G]. I show that if [G] is center-free and has trivial Schur multiplier the mod [l] monodromy will be big as long as the number of branch points of a curve in the family is chosen to be sufficiently large. Along the way the necessary algebraic results, including a generalized equivariant Witt's lemma, are presented. The proof relies on a characterization of the connected components of Hurwitz Spaces due to Ellenberg, Venkatesh, and Westerland that generalizes an older result of Conway-Parker and Fried Völklein. Connections to current results on monodromy of cyclic covers are also discussed.
On title page [l] is represented by a script "l".
On title page [G] is italicized.
Adviser: Jordan S. Ellenberg.
Ph.D. University of Wisconsin--Madison 2016.
Includes bibliographical references (pages 59-63).