Complete affine manifolds with an infinite amenable normal subgroup have amenable category at most their dimension, so all three topological volumes vanish.
Minimal volume entropy and fiber growth
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abstract
This article deals with topological assumptions under which the minimal volume entropy of a closed manifold $M$, and more generally of a finite simplicial complex $X$, vanishes or is positive. These topological conditions are expressed in terms of the growth of the fundamental group of the fibers of maps from a given finite simplicial complex $X$ to lower dimensional simplicial complexes $P$. We also give examples of finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy.
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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.