Complete affine manifolds with an infinite amenable normal subgroup have amenable category at most their dimension, so all three topological volumes vanish.
The Auslander conjecture for dimension less then 7
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abstract
In 1964 L. Auslander conjectured that every crystallographic subgroup of an the affine group is virtually solvable, i.e. contains a solvable subgroup of finite index. D. Fried and W. Goldman proved Auslander's conjecture for n = 3 using cohomological arguments. We prove the Auslander conjecture for n < 7. The proof is based mainly on dynamical arguments. In some cases, we use the cohomological argument which we could avoid but it would significantly lengthen the proof.
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Topological volumes of certain complete affine manifolds
Complete affine manifolds with an infinite amenable normal subgroup have amenable category at most their dimension, so all three topological volumes vanish.