Simplicial volume and affine manifolds

The simplicial volume is a homotopy invariant for oriented closed connected manifolds introduced by Gromov in the 80's. It is defined as the l1-seminorm of the real fundamental class of the manifold. Affine manifolds are manifolds that admit a distinguished smooth atlas with transition maps that are restrictions of affine transformations of an Euclidean space. The goal of this thesis is to study vanishing results for the simplicial volume of certain affine manifolds. We prove that the simplicial volume of an aspherical affine manifold M vanishes if the holonomy representation is injective and its image contains a non-trivial pure translation. The proof of this result, due to Bucher, Connell and Lafont, employs mainly algebraic techniques. We then investigate an alternative strategy of proof. Using topological techniques, we show that complete affine manifold whose fundamental group admits an infinite amenable normal subgroup have vanishing simplicial volume.