#
IHP program

Geometry and analysis of surface group representations

January-March 2012

##
Virtual properties of 3-manifolds

Ian Agol

4h

Concentration period "Surfaces and 3-manifolds".

We'll discuss the relations between various
group theoretic properties and properties of finite-sheeted
covers of 3-manifolds. It is a conjecture of Waldhausen
that every aspherical 3-manifold has a finite-sheeted cover
which is Haken. With the geometrization theorem, the
remaining open case is for closed hyperbolic 3-manifolds.

There are various properties of groups: LERF, large, RFRS,
virtual betti number, etc. We will discuss how work of
Wise and Kahn-Markovic implies the equivalence between
these conditions for closed hyperbolic 3-manifold groups and
the virtual Haken conjecture, and what further
implications these properties have, in particular for the
virtual fibering conjecture and the structure of the Thurston
norm. We will also consider the prospects of extending these results
to aspherical 3-manifolds with non-zero Gromov norm.