Geometry, Topology and Dynamics Seminar
Nathan Dunfield
UIUC
Integer homology 3-spheres with large injectivity radius
Abstract: Conjecturally, the amount of torsion in the first homology group
of a hyperbolic 3-manifold must grow rapidly in any exhaustive
tower of covers (see Bergeron-Venkatesh and F. Calegari-Venkatesh).
In contrast, the first betti number can stay constant (and zero)
in such covers. Here "exhaustive" means that the injectivity
radius of the covers goes to infinity. In this talk, I will explain
how to construct hyperbolic 3-manifolds with trivial first homology
where the injectivity radius is big almost everywhere by using ideas
from Kleinian groups. I will then relate this to the recent work
of Abert, Bergeron, Biringer, et. al. In particular, these examples
show a differing approximation behavior for $L^2$ torsion as compared
to $L^2$ betti numbers. This is joint work with Jeff Brock.
Monday February 18, 2013 at 3:00 PM in SEO 636