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Speaker: Mike Fried, UC Irvine

Title: Limit groups: Applying Weigel's p-Poincare duality result to label components in modular tower levels.


Modular Towers (MTs) generalize modular curve towers. Their name comes from this resemblance and from the ubiquitous appearance of modular representations (homological algebra) to decipher the tower level properties. Unlike Shimura varieties they use the more general and flexible moduli of curve covers. The advantage of this is they are suitable for a wide variety of new applications. Modular curves systematically use cusps. MT's has a group approach to those cusps that allows generalizing modular curves and their applications. This uses combinatorial groups: subgroups and quotients of braid groups and mapping class groups. These act on Nielsen classes defined by conjugacy classes in finite groups G. Applications vary with choices of conjugacy classes and with equivalences on Nielsen classes. The main MT conjecture matters only when a particular tower has a projective system of components. We rephrase finding such systems to solving embedding problems for group extensions. Here are our key words: What are the maximal p-Frattini quotients (limit groups) of orientable dimension 2 p-Poincare dual groups defined by a mapping class group orbit. We use Weigels theorem to get results on possible limit groups. Even modular curves give something new. A universal Heisenberg group obstruction shows why this case has a unique limit group. A well supported conjecture suggests when the limit group is maximal possible: equal to the full universal p-Frattini cover of G. It is when a component has a g-p' cusp. We will explain some Inverse Galois applications that use a special case of g-p' cusps, called Harbater-Mumford.

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