Vanishing of l^2-cohomology as a computational problem

TL;DR

This paper proves the undecidability of determining trivial l^2-cohomology for certain universal covers, linking it to the zero-divisor problem in group rings, and discusses conditions for decidability.

Contribution

It establishes the undecidability of l^2-cohomology triviality for specific groups and connects it to the zero-divisor problem, providing conditions for decidability under standard conjectures.

Findings

Deciding l^2-cohomology triviality is undecidable for certain groups.

No algorithm exists to determine zero-divisors in the group ring of ( extZ_2 times Z)^4.

Conditional on conjectures, such algorithms exist for groups with decidable word problem.

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.