Complex hyperbolic quasi-Fuchsian groups.

Let $Sigma$ be a closed, orientable surface of genus $gge 2$ and
let $pi_1=pi_1(Sigma)$ be its fundamental group. The different ways
of putting a hyperbolic metric on $Sigma$ are parametrised by
Teichmüller space, which we can think of as the discrete, faithful,
totally loxodromic representations of $pi_1$ to ${
m SL}(2,{mathbb R})$
up to conjugation. A huge amount is known about this space.
A classical generalisation of Teichmüller space is quasi-Fuchsian
space which is the collection of discrete, faithful, totally
loxodromic representations of $pi_1$ to ${
m SL}(2,{mathbb C})$
up to conjugation, and concerns three dimensional hyperbolic structures
on interval bundles over $Sigma$. This talk will focus on a related
space, complex hyperbolic quasi-Fuchsian space. This is the space
of discrete, faithful, totally loxodromic representations of $pi_1$
to ${
m SU}(2,1)$ up to conjugation, and concerns the complex
hyperbolic geometry of disc bundles over $Sigma$. This space is
more complicated that either Teichmüller space or the classical
quasi-Fuchsian space, and is much less well understood. The purpose of
this talk will be to survey what is currently known about this space and
what we expect to be true. We will focus on the geometric action of
complex hyperbolic representations of $pi_1$.