This semester, this seminar will focus on the study of essential surfaces in 3-manifolds and

related concepts, such as the virtual Haken conjecture, the virtual positive

first betti number conjecture, and Heegaard gradient. Principally we will

be interested in the background results for a theorem of Lackenby which

implies that hyperbolic 3-orbifolds with non-trivial vertices in the singular

locus contain immersed essential surfaces. We will discuss property (tau),

the Golod-Shafarevich property, and properties of Kleinian groups.

Participants will be expected to give a talk on one of the topics in the seminar.

Source Papers:

Surface subgroups of Kleinian groups with torsion

Heegaard genus and property (tau) for hyperbolic 3-manifolds

LERF and the Lubotzky-Sarnak conjecture

Free groups in lattices

Covering spaces of arithmetic 3-orbifolds

Covering spaces of 3-orbifolds

The asymptotic behaviour of Heegaard genus

Heegaard splittings, the virtually Haken conjecture and Property (tau)

This paper attempts to construct incompressible surfaces in finite covering spaces

of a hyperbolic 3-manifold by performing ``weak reductions" of Heegaard splittings

Large embedded balls and Heegaard genus in negative curvature

Some of the arguments in this paper (based on pleated surface interpolation

techniques going back to Thurston) may replace arguments using minimal

surfaces in the previous paper

Heegaard splittings of compact 3-manifolds

A survey paper on Heegaard splittings by Marty Scharlemann