Vanishing simplicial volume for certain affine manifolds

@article{Bucher2016VanishingSV,
title={Vanishing simplicial volume for certain affine manifolds},
author={Michelle Bucher and Chris Connell and J.-F. Lafont},
journal={arXiv: Geometric Topology},
year={2016}
}
• Published 4 October 2016
• Mathematics
• arXiv: Geometric Topology
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. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of $\pi_1$ to have vanishing simplicial volume. This answers a special case of a question due to L\"uck.
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References

SHOWING 1-10 OF 25 REFERENCES
Milnor-Wood inequalities for manifolds which are locally a product of surfaces
• Mathematics
• 2009
The classical Milnor-Wood inequalities for the Euler number of flat oriented vector bundles over surfaces are generalized for higher dimensional closed manifolds that admit a local structure of
Chern’s conjecture for special affine manifolds
An affine manifold X in the sense of differential geometry is a differentiable manifold admitting an atlas of charts with value in an affine space, with locally constant affine change of coordinates.
Elementary amenable groups and 4-manifolds with Euler characteristic 0
Abstract We extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a
Continuous Bounded Cohomology of Locally Compact Groups
The purpose of this monograph is (a) to lay the foundations for a conceptual approach to bounded cohomology; (b) to harvest the resulting applications in rigidity theory. Of central importance is the
Geometry and Topology of 3-manifolds
Low-dimensional topology is an extremely rich eld of study, with many dierent and interesting aspects. The aim of this project was to expand upon work previously done by I. R. Aitchison and J.H.
Fundamental groups of aspherical manifolds and maps of non-zero degree
We define a new class of irreducible groups, called groups not infinite-index presentable by products or not IIPP. We prove that certain aspherical manifolds with fundamental groups not IIPP do not
COLLAPSING RIEMANNIAN MANIFOLDS WHILE KEEPING THEIR CURVATURE BOUNDED . II
• Mathematics
• 2008
This is the second of two papers concerned with the situation in which the injectivity radius at certain points of a riemannian manifold is "small" compared to the curvature. In Part I [3], we
ON THE HOMOLOGICAL DIMENSION OF GROUP ALGEBRAS OF SOLVABLE GROUPS
In the paper we calculate the weak dimension of the group algebra of a solvable group and the projective dimension of the group algebra of a countable nilpotent group. Exact bounds are obtained for
The radiance obstruction and parallel forms on affine manifolds
• Mathematics
• 1984
A manifold M is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine. When they are locally linear M is called radiant. The obstruction to radiance is a