The groups in question are what we referred to last week as "extreme" extensions of finite groups. They are Frattini covers: special homomorphisms of one finite group to another that are quotients of a certain universal extension (of any finite group).
The towers of varieties they bind include classical towers of what are called modular curves. By generalizing the properties of those modular curve towers we connect two books of J.P. Serre: "Abelian l-adic Representations" (from elliptic curves) and "Topics in Galois Theory" (about the regular Inverse Galois Problem). These books were written over 25 years apart.
We will construct these group covers from two viewpoints, structurally and homologically. This will allow understanding the structure of the spaces they produce, for which our examples will be upper half plane quotients by finite index subgroups of PSL_2(Z). A very small example of such a cover is the Spin cover of Alternating groups.
In this talk, I will first introduce Jonathan Rosenberg’s notion of a Levi-Civita connection for a Riemannian metric on a rank- free Hilbert module over a ‘generically transcendental’ quantum torus of dimension . I will then describe how to use Rosenberg’s Levi-Civita connections to construct metrized quantum vector bundles over ‘generically transcendental’ quantum tori, thus providing another class of examples for Frédéric Latrémolière’s work on the modular Gromov-Hausdorff propinquity. A technical tool that appears in the construction is the Minkowski gauge functional, which was used by Marc Rieffel in his work on Leibniz seminorms. (The new examples presented in this talk constitute the first part of ongoing joint work with Konrad Aguilar.)
Constructing Metrized Quantum Vector Bundles over Quantum Tori from Rosenberg’s Levi-Civita Connections