12/8/2005

Nathan Dunfield
Caltech

Hyperbolic rational homology 3-spheres of large injectivity radius

(Joint work with Frank Calegari, Harvard) I will discuss a question of Cooper adjacent to the Virtual Haken Conjecture.  In particular, the point of the talk is that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These examples come from a tower of abelian covers of an explicit arithmetic 3-manifold.   Initially, Frank Calegari and I investigated these examples using automorphic forms, and we needed to to assume are the Generalized Riemann Hypothesis and a mild strengthening of results of Taylor et al. on part of the Langlands Program for GL_2 of an imaginary quadratic field.  However, an alternate approach of Nigel Boston and Jordan Ellenberg using pro-p groups which gives the result unconditionally.  This talk will focus on a slight variation of the Boston-Ellenberg approach that minimizes the p-groups machinery involved to a single non-trivial theorem.