## Complex hyperbolic quasi-Fuchsian groups. [1]

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 $pi_1$.