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).
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?
Theorem 2 Let be an aspherical affine manifold. Assume that
- the holonomy representation is faithful,
- its image contains a pure translation.
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 3 This is true if the map
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 .