In analogy with ordinary simplicial volume, we show that integral foliated simplicial volume of oriented closed connected aspherical n-manifolds that admit an open amenable cover of multiplicity at most n is zero. This implies that the fundamental groups of such manifolds have fixed price and are cheap as well as reproves some statements about homology growth.

This work was supported by the CRC 1085 Higher Invariants (Universität Regensburg, funded by the DFG) and by the RTG 2229 Asymptotic Invariants and Limits of Groups and Spaces (KIT, funded by the DFG).