** Vanishing simplicial volume for certain affine manifolds **

Joint work with C. Connell and J.-F. Lafont.

**1. Affine manifolds **

An affine manifold is a compact smooth -manifold equipped with an atlas of maps to with coordinate changes in the affine group . An affine manifold has a developing map . An affine manifold is complete if every affine segment can be extended forever. Equivalently, if the developing map is 1-1 (the is a quotient of by a discrete group of affine transformations).

** 1.1. Examples **

Quotients, like the circle .

Mapping tori of affine toral automorphisms.

Products of hyperbolic manifolds with the circle (modelled on cone , where is the pseudosphere in .

** 1.2. Famous problems **

**Auslander conjecture**. The fundamental group of a complete affine manifold is virtually solvable.

Known in low dimensions (Fried-Goldman if , Abels-Margulis-Soifer for ).

**Chern conjecture**. The Euler characteristic of every affine manifold vanishes.

Known in 2 dimensions (Benzecri), in the complete case (Koslark-Sullivan), when there is an invariant volume (Klingler), for irreducible higher rank locally symmetric spaces (Margulis), local products of surfaces (Bucher-Gelander), connected sums of manifolds with finite fundamental groups (Smillie).

**Markus conjecture**. An affine manifold is complete iff it has an invariant volume.

**2. Simplicial volume **

This is the norm of the fundamental class (Gromov).

For every continuous map between -manifolds,

hence for tori, for instance. On the other hand, for a higher genus surface, . More generally, for a hyperbolic -manifold, , where is the volume of the regular ideal simplex in hyperbolic -space (Gromov-Thurston). Similar formulae are conjectured for other locally symmetric spaces, but the only known cases is for quotients of (Bucher).

Theorem 1 (Sullivan-Smillie)For affine -manifolds, .

In fact, all what is used is the flatness of the tangent bundle. The proof constructs a PL vectorfield with at most one zero per simplex of a triangulation.

This suggests a **strengthening of Chern’s conjecture**: does for all affine manifolds?

** 2.1. Result **

Theorem 2Let be an aspherical affine manifold. Assume that

- the holonomy representation is faithful,
- its image contains a pure translation.

Then .

** 2.2. Proof **

Lueck asked wether every aspherical manifold whose fundamental group has a nontrivial normal amenable subgroup has vanishing simplicial volume. We give a partial answer.

Lemma 3This is true if the map

vanishes.

The point is to show that the map is not onto. Amenability implies that is an isomorphism. A commutative diagram show that vanishes.

We apply the Lemma to the kernel of linear holonomy , which by assumption, maps injectively to translations of .