# Colloquium abstracts

**Ludovic Marquis**

l'Ecole Normale Superieure de Lyon, France

February 11, 2010

**Convex Projective Manifold of finite volume**: In my talk, I will explain how convex projective geometry is a generalisation of hyperbolic geometry.

A convex projective manifold M is the quotient of a properly open convex Omega set by a discrete group of projective transformation G. The basic example of such manifold is the quotient of the hyperbolic space by a discrete group of isometries.

This kind of manifold carry a natural measure. A lot of people have studied the case where the manifold M is compact. I will explain what is known when the dimension of M is 2 and how to construct such a manifold when Omega is not the hyperbolic space.

This will lead us, to the construction of discrete subgroup of SL(n+1,R) which are Zariski dense but not lattice of SL(n+1,R).