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We show that it is impossible to algorithmically decide if the    l  2   ‐cohomology of the universal cover of a finite CW‐complex is trivial, even if we only consider complexes whose fundamental group is equal to the elementary amenable group(    Z  2≀Z)   3   . A corollary of the proof is that there is no algorithm which decides if an element of the integral group ring of the group(    Z  2≀Z)   4    is a zero‐divisor. On the other hand, we show, assuming some standard conjectures, that such an algorithm exists for the integral group ring of any group with a decidable word problem and a bound on the sizes of finite subgroups.

Vanishing of l 2 ‐cohomology as a computational problem