Vanishing of l2-cohomology as a computational problem

Grabowski, Łukasz (2015) Vanishing of l2-cohomology as a computational problem. Bulletin of the London Mathematical Society, 47 (2). pp. 233-247. ISSN 0024-6093

Abstract

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.