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Vanishing of l2-cohomology as a computational problem

Research output: Contribution to Journal/MagazineJournal articlepeer-review

<mark>Journal publication date</mark>1/04/2015
<mark>Journal</mark>Bulletin of the London Mathematical Society
Issue number2
Number of pages15
Pages (from-to)233-247
Publication StatusPublished
<mark>Original language</mark>English


We show that it is impossible to algorithmically decide if the l2-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 (Z2≀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 (Z2≀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.