TY - JOUR

T1 - Vanishing of l2-cohomology as a computational problem

AU - Grabowski, Łukasz

PY - 2015/4/1

Y1 - 2015/4/1

N2 - 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.

AB - 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.

U2 - 10.1112/blms/bdu114

DO - 10.1112/blms/bdu114

M3 - Journal article

VL - 47

SP - 233

EP - 247

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

IS - 2

ER -