Connected Components of Sphere Cover Families of An-type

Our basic question: Restricting to covers of the sphere by a compact Riemann surface of a given type, do all such compose one connected family? Or failing that, do they fall into easily discerned components? The answer has often been “Yes!,” figuring in the connectedness of the moduli space of curves of genus g (geometry), Davenport’s problem (arithmetic) and the genus 0 problem (group theory). One consequence is that we then know the definition field of the family components. Our main goal is to explicitly describe specific projective sequences of such families, called M(odular) T(ower)s. This shows precisely why the Main MT Conjecture holds: high tower levels have general type and, for K a fixed number field, no K points. We start with connectedness results for certain absolute Hurwitz spaces – examples of Liu-Osserman – of alterating group covers. The inner versions of these spaces are level 0 of our MTs. Connectedness results ensure certain cusp types – especially those defined by the shift of a H(arbater)-M(umford) representative – lie on a tower level boundary. Another type, a p-cusp, directly contributes to showing the Main MT Conjecture. Modular curve towers have both pand H-M related cusps, and no others. General MTs, can have another cusp type. This is like our examples, where p = 2, which have no p-cusps at level 0. Still, this 3rd type often disappears at higher levels, to be replaced by p-cusps. Our cusp description uses modular representations, rather than semi-simple representations. The sh-incidence matrix, from a natural pairing on cusps, simplifies displaying results. A lift invariant explains the nature of both components and cusps.