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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.
Abstract:
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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