## Notes of Michelle Bucher’s Cambridge lecture 22-06-2017

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 ${n}$-manifold equipped with an atlas of maps to ${{\mathbb R}^n}$ with coordinate changes in the affine group ${{\mathbb R}^n\times Gl(n,{\mathbb R})}$. An affine manifold has a developing map ${\tilde M\rightarrow{\mathbb R}^n}$. An affine manifold is complete if every affine segment can be extended forever. Equivalently, if the developing map is 1-1 (the ${M}$ is a quotient of ${{\mathbb R}^n}$ by a discrete group of affine transformations).

1.1. Examples

Quotients, like the circle ${S^1={\mathbb R}/{\mathbb Z}}$.

Mapping tori of affine toral automorphisms.

Products of hyperbolic manifolds with the circle (modelled on cone ${H^n\times{\mathbb R}}$, where ${H^n}$ is the pseudosphere in ${{\mathbb R}^{n,1}}$.

1.2. Famous problems

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

Known in low dimensions (Fried-Goldman if ${n=3}$, Abels-Margulis-Soifer for ${n=4,5,6}$).

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 ${\ell^1}$ norm of the fundamental class (Gromov).

For every continuous map ${f:M\rightarrow N}$ between ${n}$-manifolds,

$\displaystyle \begin{array}{rcl} |deg(f)|\|M\|\leq \|N\|, \end{array}$

hence ${\|M\|=0}$ for tori, for instance. On the other hand, for a higher genus surface, ${\|M\|=-2\chi(M)}$. More generally, for a hyperbolic ${n}$-manifold, ${vol(M)=v_n\|M\|}$, where ${v_n}$ is the volume of the regular ideal simplex in hyperbolic ${n}$-space (Gromov-Thurston). Similar formulae are conjectured for other locally symmetric spaces, but the only known cases is ${\|M\|=6\chi(M)}$ for quotients of ${H^2\times H^2}$ (Bucher).

Theorem 1 (Sullivan-Smillie) For affine ${n}$-manifolds, ${|\chi(M)|\leq 2^{-n}\|M\|}$.

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 ${\|M\|=0}$ for all affine manifolds?

2.1. Result

Theorem 2 Let ${M}$ be an aspherical affine manifold. Assume that

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

Then ${\|M\|=0}$.

2.2. Proof

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

Lemma 3 This is true if the map

$\displaystyle \begin{array}{rcl} H^n(\pi_1(M)/A)\rightarrow H^n(\pi_1(M)) \end{array}$

vanishes.

The point is to show that the map ${H_b^n(M)\rightarrow H^n(M)}$ is not onto. Amenability implies that ${H^n(\pi_1(M)/A)\rightarrow H_b^n(\pi_1(M))}$ is an isomorphism. A commutative diagram show that ${H_b^n(M)\rightarrow H^n(M)}$ vanishes.

We apply the Lemma to the kernel of linear holonomy ${A=ker(\pi_1(M)\rightarrow Gl(n,{\mathbb R}))}$, which by assumption, maps injectively to translations of ${{\mathbb R}^n}$.