Quantitative connections between hyperbolic volume and topological complexity

Marc Culler, University of Illinois at Chicago

We will discuss joint work with Ian Agol and Peter Shalen in which we
obtain lower bounds on the volume of a hyperbolic manifold that
exhibits a certain degree of topological complexity.  For example, if
a hyperbolic manifold has mod 5 first homology of rank greater than 2
then it must have volume at least 1.22.  The methods combine
displacement estimates based on Patterson-Sullivan conformal
densities, topological embedding theorems proved with towers of
coverings, and other recent results such as the Marden Tameness
Conjecture, proved by Agol and Calegari-Gabai, and the bounds on
volume change under drilling due to Agol, Storm, Thurston and
Dunfield.