Description:
<p>We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. As a consequence, these manifolds have zero Euler characteristic, satisfying the Chern Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1">
<mml:semantics>
<mml:msub>
<mml:mi>π<!-- π --></mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:annotation encoding="application/x-tex">\pi _1</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> to have vanishing simplicial volume. This answers a special case of a question due to Lück.</p>