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}.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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