Vanishing of l^2-cohomology as a computational problem
arXiv:1202.1162 · doi:10.1112/blms/bdu114
Abstract
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 \wr 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 \wr 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.
18 pages; rewritten following referee's reports; to appear in Bulletin of LMS