Title: Finite group theory and Connectedness of moduli spaces of Riemann Surface covers

Abstract: Connections between the R(egular)I(nverse)G(alois)P(roblem) and the S(trong)T(orsion)C(onjecture) on abelian varieties arise from inspecting properties of Hurwitz spaces: families of sphere covers of a specific type. We start with two results:

1. The "3-cycles result" on connected components of Hurwitz spaces r 3-cycle covers: If the covers have degree n and r=n-1, then there is one component; if r ≥ n, there are two components. Describing these components is a warm up on the Fried-Serre Spin-Lifting invariant.

2. The Liu-Osserman connectedness result on genus 0 pure-cycle Hurwitz spaces: For covers with just one ramified point over each branch point, the unordered ramification orders determine a unique component.

I will list "cusp" properties of certain spaces appearing in #1 and #2. Above each space I will construct a tower of spaces (the M(odular) T(ower)) attached to a prime p. All MT levels will be curves and upper-half plane quotients (j-line covers), though not modular curves. The connection between the RIGP and STC appears in a conjectured property of high tower levels:

(*) like the modular curve Y(p^k+1) for k large, high MT levels should have no rational points.

A fundamental result shows (*) follows if high MT levels have a "p cusp." An action from a braid group on 4 strings interprets this. I will use the Spin-lifting invariant to prove -- often -- there is a "p cusp." The talk directs attention to the role of group theory, especially of Schur multipliers, in describing families of Riemann surfaces. I conclude with a conjecture that simultaneously generalizes #1 and #2.