  A smooth five-dimensional s-cobordism becomes a smooth product if stabilized
by a finite number n of $S^2xS^2x[0,1]$'s. We show that for amenable
fundamental groups, the minimal n is subextensive in covers, i.e.,
n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width,
which is a bridge between 4-dimensional topology and coarse geometry.
