Abstract
Simplicial volumes measure the complexity of fundamental cycles of manifolds. In this article, we consider the relation between the simplicial volume and two of its variants — the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space.
We show that the integral foliated simplicial volume is monotone with respect to weak containment of measure preserving actions and yields upper bounds on (integral) homology growth.
Using ergodic theory we prove that the simplicial volume, integral foliated simplicial volume and stable integral simplicial volume coincide for closed hyperbolic 3-manifolds and closed aspherical manifolds with an amenable residually finite fundamental group (being equal to zero in the latter case).
However, we show that the integral foliated simplicial volume and the classical simplicial volume do not coincide for hyperbolic manifolds of dimension at least 4.
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Frigerio, R., Löh, C., Pagliantini, C. et al. Integral foliated simplicial volume of aspherical manifolds. Isr. J. Math. 216, 707–751 (2016). https://doi.org/10.1007/s11856-016-1425-3
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DOI: https://doi.org/10.1007/s11856-016-1425-3