Big mod ℓ monodromy for families of G covers.

摘要

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 Volklein. Connections to current results on monodromy of cyclic covers are also discussed.