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# Complex hyperbolic quasi-Fuchsian groups.

Jeudi, 10 Novembre, 2005 - 15:00
Prénom de l'orateur :
John
Nom de l'orateur :
PARKER
Résumé :

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 ${ m SL}(2,{mathbb R})$
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
complex hyperbolic representations of $pi_1$.