Growth of Betti numbers

We discuss growth rates of Betti numbers in a family of coverings of a compact cell complex X, when the corresponding L2 Betti number of X is zero. We show that the Betti numbers are bounded by a function, sub-linear in the order of the covering. If the appropriate Novikov–Shubin invariant of X is positive, the rate bounds are improved. For well behaved families (such as congruence covers of symmetric spaces), if the L2 spectrum of X̃ has a gap at zero then the growth rate is bounded by the order of the covering raised to a power less than one.